UC-NI 


LOGIC 


OUTLINES  OF  LOGIC. 


BY 

JACOB  WESTLTJND, 
\t 

PROFESSOR   OF    MATHEMATICS   IN    BETHANY   COLLEGE, 

LINDSBOBG,  KANSAS. 


TOPEKA,    KANSAS: 

CRANE    &    CO.,    PUBLISHERS. 

1896. 


Copyright,  1896,   by  CRANE   &  Co. 


TABLE    OF    COIsrTEJSrTS. 


INTRODUCTION. 

DEFINITION   AND    SCOPE    OF   LOGIC. 

PAGE. 

1.  DEFINITION  OF  LOGIC 1 

2.  UTILITY  OF  LOGIC 2 

3.  OPERATIONS  OF  THE  MIND 2 

CHAPTER  I. 
FUNDAMENTAL  LAWS  OF  THOUGHT. 

1.  DEFINITION  OF  LAW  OF  THOUGHT 4 

2.  FUNDAMENTAL  LAWS  OF  THOUGHT 4 

1.  Law  of  Identity 4 

2.  Law  of  Contradiction 5 

3.  Law  of  Excluded  Middle 5 

4.  Law  of  Sufficient  Reason 5 

CHAPTER  II. 

CONCEPTS. 

1.  DEFINITION  OF  CONCEPT 7 

2.  CLASSIFICATION  OF  CONCEPTS 7 

3.  CONTENT  AND  EXTENT 8 

4.  RELATION  OF  CONCEPTS: 

1.  Compatible  Concepts 10 

2.  Incompatible  Concepts 10 

3.  Subordinate  Concepts 11 

4.  Coordinate  Concepts 12 

(iii) 


215271 


OUTLINES  OF  LOGIC. 


CHAPTER  III. 

JUDGMENTS. 

PAGE. 

1.  DEFINITION  OF  JUDGMENT  ....................  .  .............   14 

2.  TERMS  .....................................................   14 

3.  CLASSIFICATION  OF  JUDGMENTS: 

1.  Quality  ................................................   15 

2.  Quantity  ...............................................   15 

3.  Relation  ..............................................  ,.    17 

4.  Modality  ..........................  ........    ............   19 

4.  DISTRIBUTION  OF  TERMS  ....................................  20 

5.  IMMEDIATE  INFERENCE  : 

1.  Synonymous  Judgments  .................................   23 

2.  Subalterns  ..............................................   24 

3.  Opposition  ............................................   25 

4.  Conversion  .....  ...................................    ....   29 

6.  SIMPLE  AND  COMPLEX  PROPOSITIONS  ........................  33 

CHAPTER  IV. 

SYLLOGISMS. 

1.  DEFINITION  OF  SYLLOGISM  ..................................  35 

2.  CLASSIFICATION  OF  SYLLOGISMS  ............................   36 

3.  CATEGORICAL  SYLLOGISMS  : 

1.  Definition  ..............................................   37 

2.  Rules  .................................................   37 

3.  Explanation  of  Rules  ....................................   38 

4.  Figures  ................................................   42 

5.  Moods  .................................................   43 

4.  HYPOTHETICAL  SYLLOGISMS: 

1.  Definition  ........  ......................................  49 

2.  Moods  .................................................  50 

3.  Fallacies  ..............................................  52 

4.  Reduction  of  Hypothetical  to  Categorical  Syllogisms  ........  53 

5.  DISJUNCTIVE  SYLLOGISMS  : 

1.  Definition  ..............................................   54 

2.  Rules  ..................................................   54 

3.  Moods  .  .  .54 


TABLE  OF  CONTENTS. 


PAGE. 

6.  DILEMMA 55 

7.  COMPOUND  SYLLOGISMS 56 

8.  ABRIDGED  SYLLOGISMS  : 

1.  Enthymeme 58 

2.  Epichirema 59 

3.  Sorites 59 

CHAPTER  V. 

FALLACIES. 

1.  DEFINITION  OF  FALLACY 63 

2.  CLASSIFICATION  OF  FALLACIES 63 

3.  LOGICAL  FALLACIES  : 

1.  Fallacy  of  Equivocation 65 

2.  Fallacy  of  Composition 66 

3.  Fallacy  of  Division 66 

4.  Fallacy  of  Accident 67 

5.  Converse  Fallacy  of  Accident 67 

6.  Fallacy  of  Many  Questions 67 

7.  Fallacy  of  Amphibology 68 

8.  Fallacy  of  Positive  and  Negative  Intention 68 

4.  MATERIAL  FALLACIES: 

1.  Begging  the  Question 69 

2.  Fallacy  of  False  Cause 69 

3.  Fallacy  of  Irrelevant  Conclusion 70 

5.  PARALOGISMS  AND  SOPHISMS 70 

CHAPTER  VI. 

METHOD. 

1.  SCIENCE  : 

1.  Definition  of  Science 72 

2.  Requisites  of  a  Science 72 

3.  Axioms 73 

2.  DEDUCTION  AND  INDUCTION  : 

1.  Definition  of  Method 73 

2.  Deduction 74 

3.  Induction  . .  .74 


vi  OUTLfNRS  OF  LOGIC. 

3.  DEFINITION:  PAGE. 

1.  Definition  defined 76 

2.  Rules  for  Definition 77 

3.  Nominal  and  Real  Definitions 80 

4.  Description 80 

4.  DIVISION: 

1.  Division  defined 81 

2.  Dichotomy 81 

3.  Rules  for  Division 82 

4.  Partition ,.  84 

5.  DEMONSTRATION  : 

1.  Demonstration  defined 85 

2.  Rules  for  Demonstration 85 

3.  Classification  of  Demonstrations 86 

6.  ANALOGY 90 

7.  HYPOTHESIS 91 

8.  CLASSIFICATION  OF  SCIENCES..  .  91 


EXERCISES 92-102 


INTEODTJCTIOW. 


DEFINITION    AND    SCOPE    OF   LOGIC. 
1.   Definition  of  Logic. 

Logic  is  the  science  of  the  formal  laws  of  human 
thought. 

Logic  is  the  science  which  has  for  its  object  to  inves- 
tigate the  laws  of  human  thought  apart  from  the  other  acts 
of  the  mind.  It  explains  the  laws  and  principles  by  which 
all  reasoning  must  be  governed.  In  all  sciences  the  rea- 
soning must  be  in  accordance  with  the  principles  of  logic, 
and  although  the  method  may  be  different  in  different 
sciences  it  must  always  conform  to  the  laws  of  thought. 

Logic  is  mainly  a  formal  science,  having  for  its  object 
to  ascertain  and  describe  all  the  general  forms  in  which 
thought  presents  itself  without  regard  to  any  subject-mat- 
ter. Logic  differs  from  psychology  in  having  for  its  ob- 
ject only  the  investigation  of  the  formal  laws  of  thought, 
while  psychology  treats  of  all  the  facts  of  the  human  mind 
and  the  laws  by  which  its  operations  are  guided. 


OUTLINES  OF  LOGIC. 


2.  Utility  of  Logic. 

Logic  does  not  teach  us  to  think,  but  teaches  us  the 
laws  by  which  our  reasoning  must  be  guided.  All  per- 
sons learn  to  think  and  to  reason  even  before  they  know 
the  name  of  logic,  and  thus  unconsciously  apply  the  prin- 
ciples of  logic ;  but  many  questions  are  of  so  complex  and 
difficult  a  nature  that  it  is  only  by  the  aid  of  logic  that  we 
are  able  to  detect  what  is  correct  or  fallacious  in  the  argu- 
ment. The  chief  utility  of  logic  thus  consists  in  giving  an 
invariable  test  of  the  correctness  of  an  argument. 

3.  Operations  of  the  Mind. 

In  approaching  an  argument  the  mind  passes  through 
the  following  intellectual  processes:  Perception,  abstrac- 
tion, generalization,  judgment,  and  reasoning. 

1.  Perception  is  the  act  of  the  mind  by  which  it  gains 
knowledge  of  external  objects  through  the  senses. 

The  products  of  perception  are  called  percepts.  Thus 
my  idea  of  my  house,  or  of  Boston,  or  of  any  particular 
object,  is  a  percept. 

2.  Abstraction  is  the  act  of  the  mind  by  which  it  draws 
a  quality  away  from  an  object  and  considers  it  apart  from 
the  other  peculiarities  of  the  object. 

Thus  the  observing  of  the  color  of  a  certain  object  and 
making  that  a  distinct  object  of  thought  to  the  exclusion 


INTRODUCTION. 


of  all  the  other  qualities  of  the  same  object,  is  a  process 
of  abstraction. 

3.  Generalization  is  the  act  of  the  mind  by  which  it  con- 
siders the  qualities  ivhich  are  common  to  all  the  individuals 
of  a  group  of  objects  and  unites  them  into  a  single  notion 
comprehending  them. 

Thus  if  we  consider  the  properties  common  to  all  kinds 
of  triangles,  disregarding  difference  in  size  or  shape,  the 
process  is  generalization. 

A  concept  or  general  notion  is  the  product  of  abstraction 
and  generalization. 

The  concept  plant,  for  instance,  is  formed  by  fixing  our 
attention  upon  the  properties  common  to  all  individual 
plants  and  disregarding  all  the  points  in  which  they  differ. 
The  concept  plant  thus  embraces  all  individual  plants,  and 
is  a  name  that  may  be  applied  to  any  one  of  them. 

A  concept  is  always  general,  a  percept  particular. 

4.  Judgment  is  the  act  of  the  mind  by  which  we  com- 
pare two  objects  of  thought,  asserting  whether  they  agree 
or  not. 

The  product  of  this  operation  is  called  a  judgment.  A 
judgment  expressed  in  words  is  called  &  proposition. 

5.  Reasoning  is  the  act  of  the  mind  which  consists  in 
drawing  conclusions  from  two  or  more  judgments. 

An  act  of  reasoning  in  its  simplest  logical  form  is  called 


CHAPTER  I. 


FUNDAMENTAL   LAWS   OF   THOUGHT. 

1.   Definition  of  Law  of  Thought. 

A  law  of  thought  is  a  necessary  and  universal  principle 
by  which  all  thought  must  be  governed. 

2.   Fundamental  Laws  of  Thought. 

There  are  four  fundamental  laws  of  thought,  on  which 
all  reasoning  must  ultimately  depend.  These  laws  are : 

1.  The  Law  of  Identity  (Principium  iderititatis). 

2.  The  Law  of  Contradiction  (Principium  contradictionis). 

3.  The  Law  of  Excluded  Middle  (Principium  exclusi  tertii). 

4.  The   Law   of  Sufficient   Reason    (Principium    rationis 
sufficients). 

1.  THE  LAW  OF  IDENTITY. —  Whatever  is,  is. 

This  law  may  be  expressed  by  the  formula  A=A.  Its 
meaning  is  that  everything  is  identical  with  itself.  All 
the  attributes  of  a  thing  must  be  consistent  with  each 
other  and  with  the  thing  itself.  In  the  proposition  All 

(4) 


FUNDAMENTAL  LAWS  OF  THOUGHT. 


Americans  are  rational  beings,  the  identity  of  All  Ameri- 
cans with  some  rational  beings  is  set  forth. 

2.  THE  LAW  OF  CONTRADICTION. — Nothing  can  both,  be 
and  not  be. 

This  law  may  be  expressed  by  the  formula  A  not  = 
not — A.  The  attributes  of  an  object  must  not  be  incon- 
sistent with  each  other  nor  with  the  thing  itself.  In  the 
proposition,  No  animals  are  plants,  we  assert  that  ani- 
mals are  inconsistent  with  plants.  A  triangle  may  be 
either  right-angled  or  not  right-angled,  but  we  cannot 
conceive  that  it  should  be  both  at  the  same  time.  If  we 
say  that  a  triangle  is  round,  we  evidently  violate  this  law. 
because  roundness  is  a  quality  inconsistent  with  a  triangle. 

3.  THE  LAW  OF  EXCLUDED  MIDDLE. — Everything  must 
either  be  or  not  be. 

This  law  may  be  expressed  by  the  formula,  A  is  either 
B  or  not — B.  It  is  impossible  to  conceive  of  any  thing 
and  any  quality  without  affirming  that  the  quality  either 
belongs  to  the  thing  or  does  not  belong  to  it.  Gold,  for 
instance,  must  be  either  a  metal  or  not  a  metal.  There  is 
no  third. 

4.  THE  LAW  OF  SUFFICIENT  REASON. — For  every  conse- 
quent there  must  be  a  sufficient  reason. 

If  two  propositions  are  connected  in  such  a  manner 
that  the  truth  of  one  necessarily  implies  the  truth  of  the 


OUTLINES  OF  LOG'IC. 


other,  the  former  is  called  the  reason  and  the  latter  the 
consequent. 

This  law  may  be  expressed  by  the  formula,  If  A  is,  B 
is.  Its  meaning  is,  that  for  every  proposition  that  is  not 
intuitively  true  a  sufficient  reason  must  be  given. 

For  instance,  If  two  triangles  have  equal  bases  and 
equal  altitudes,  they  are  equivalent.  Here  the  equivalence 
of  the  triangles  is  the  consequent,  and  the  reason  why 
they  are  equivalent  is  that  they  have  equal  bases  and 
equal  altitudes. 


CHAPTER  II. 


CONCEPTS. 

1.  Definition  of  Concept. 

A  concept  or  general  notion  is  the  consciousness  in  our 
mind  of  the  attributes  common  to  all  the  individuals  of  a 
certain  group  of  objects. 

Concepts  are  formed  by  abstraction  and  generalization, 
as  has  already  been  mentioned.  As  examples  of  concepts 
we  may  give  the  following :  Man,  animal,  book,  triangle, 
plant,  planet,  heavenly  body,  and  dog. 

2.  Classification  of  Concepts. 

Concepts  may  be  divided  into  1.  Positive  and  negative  / 
2.  Absolute  and  Relative /  3.  Concrete  and  Abstract. 

1.  a)  A  positive  concept  is  one  in  which  the  existence 
of  a  quality  is  asserted. 

b)  A  negative  concept  is  one  in  which  the  absence  of 
a  quality  is  asserted. 

(7) 


OUTLINES  OF  LOGIC. 


Thus,  organic  and  right-angled  are  positive  and  inor- 
ganic and  not  right-angled  negative  concepts. 

2.  d)  An  absolute  concept  is  one  that  can  be  thought  of 
without  reference  to  some  other  concept. 

b)  A  relative  concept  is  one  that  cannot  be  thought 
of  without  reference  to  some  other  concept. 

Thus,  father,  mother,  son,  and  daughter  are  relative  con- 
cepts. We  cannot  think  of  father  or  mother  without  ref- 
erence to  a  child,  nor  of  son  or  daughter  without  reference 
to  father  or  mother.  Metal,  water,  and  triangle,  on  the 
other  hand,  are  terms  which  have  no  apparent  relation  to 
any  other  things,  and  which  therefore  are  absolute. 

3.  a)  A  concrete  concept  is  a  name  that  can  be  applied 
to  a  thing. 

b)  An  abstract  concept  is  the  name  of  a  quality  that 
belongs  to  a  thing. 

Thus  circle,  table,  and  brick-house  are  concrete ;  but 
redness,  hardness,  and  usefulness  abstract  concepts. 

3.  Content  and  Extent. 

Every  concept  has  content  and  extent. 

By  the  content  of  a  concept  is  meant  all  the  marks  or 
attributes  of  the  concept. 

By  the  extent  of  a  concept  is  meant  all  the  individuals 
or  objects  it  embraces. 

Let  us  take  the  concept  insect.     The  content  of  insect 


CONCEPTS.  9 


consists  of  all  the  attributes  which  are  necessarily  pos- 
sessed by  all  insects  and  by  which  an  insect  is  distin- 
guished from  all  other  beings.  By  the  extent  of  insect  we 
mean  all  the  different  kinds  of  insects  that  exist. 

When  we  compare  two  concepts  that  are  related  to  one 
another,  we  observe  that  the  concept  which  is  poorer  in 
content  has  the  greater  extent,  and  that  the  one  that  has 
the  greater  content  has  the  smaller  extent ;  or  as  it  is 
usually  expressed : 

The  content  am.d  extent  of  two  concepts  are  in  inverse 
ratio  to  each  other. 

In  order  to  make  this  clear  let  us  compare  the  two  terms 
fish  and  vertebrate.  The  term  vertebrate  includes  not  only 
all  the  animals  that  are  included  under  the  term  fish,  but 
also  reptiles,  birds,  mammals,  etc.  Consequently  vertebrate 
has  a  greater  extent  than^A.  On  the  other  hand,  all  the 
properties  that  belong  to  vertebrates  must  necessarily  be- 
long to  all  fishes,  and  in  addition  to  these  there  are  many 
properties  that  belong  exclusively  to  fishes  and  by  which 
fishes  are  distinguished  from  all  other  vertebrates.  There- 
fore fish,  having  a  greater  number  of  marks  or  attributes 
than  vertebrate,  has  the  greater  content.  Vertebrate  is  a 
term  that  may  be  applied  to  ail  fishes,  and  fish  is  an  indi- 
vidual case  of  vertebrate.  As  another  example  let  us  take 
the  two  terms  plane  figure  and  circle.  Of  these  the  former 
has  obviously  the  greater  extent  and  the  latter  the  greater 
content. 


10  OUTLINES  OF  LOGIC. 

If  two  concepts  are  so  related  that  one  includes  the 
other,  as  the  concepts  vertebrate  and  fish,  the  one  that  in- 
cludes the  other  is  called  the  higher  concept,  and  the  one 
that  is  included  in  the  other  is  called  the  lower  concept. 
Thus  vertebrate  is  the  higher  and  fish  the  lower  concept. 

4.  Relation  of  Concepts. 

Two  or  more  concepts  may  be  compared :  1st,  with  re- 
spect to  content ;  and  2d,  with  respect  to  extent.  In  the 
first  case  they  may  be  either  compatible  or  incompatible. 
In  the  second  case  they  may  be  either  subordinate  or  co- 
ordinate. 

1.  COMPATIBLE  CONCEPTS. — Two  concepts  are  said  to  be 
compatible  when  they  both  can  be  affirmed  of  the  same 
subject,  or  both  are  included  in  the  content  of  the  same 
concept. 

The  two  terms  equilateral  and  right-angled,  for  instance, 
may  both  be  affirmed  of  a  square,  and  are  consequently 
compatible.  Large  and  heavy  are  also  two  compatible 
terms,  because  they  may  be  affirmed  of  the  same  subject. 

2.  INCOMPATIBLE  CONCEPTS. — Two  concepts  are  said  to 
he  incompatible  when  they  cannot  both  be  affirmed  of  the 
same  subject,  or  are  not  included  in  the  content  of  the  same 
concept. 

Incompatible  concepts  are  either  contradictory  or  con- 
trary. 


CONCEPTS.  11 


a)  Two  concepts  are  contradictory  when  one  is  the  nega- 
tive of  the  other. 

For  instance,  cold  and  not-cold,  figure  and  not-figure, 
organic  and  inorganic,  etc.  From  a  logical  point  of  view, 
it  is  immaterial  which  one  of  two  contradictory  terms  is 
considered  positive  and  which  negative.  Each  is  the  neg- 
ative of  the  other. 

5)  Two  concepts  are  contrary  when  one  not  only  implies 
a  negation  of  the  other,  hut  also  expresses  some  positive 
attribute. 

For  instance,  man  and  woman,  pentagonal  and  hexago- 
nal. 

3.  SUBORDINATE  CONCEPTS. —  Of  two  concepts,  one  is  said 
to  he  subordinate  to  the  other  when  it  is  included  in  the 
extent  of  the  other. 

If  one  concept  is  included  in  the  extent  of  another,  the 
former  is  called  species  and  the  latter  genus.  Thus,  of  the 
two  terms  plant  and  tree,  plant  is  the  genus,  and  tree  is  a 
species  of  the  genus  plant.  The  genus  has  always  a 
greater  extent  than  the  species,  i.  e.,  includes  a  greater 
number  of  individuals  than  the  species.  But  as  extent 
and  content  are  in  inverse  ratio  to  each  other,  it  follows 
that  the  species  has  greater  content  or  a  greater  number 
of  attributes  than  the  genus.  The  species  has  not  only  all 
the  attributes  of  the  genus,  but  also  other  attributes  by 
which  it  is  distinguished  from  all  other  species  of  the  same 


12  OUTLINES  OF  LOGIC. 

genus.  The  relation  between  two  concepts  of  which  one 
is  subordinate  to  the  other  is  shown  by  the  diagram,  where 
A  (the  outer  circle)  represents  the  genus  and  B  (the  inner 
circle)  the  species. 


4.  COORDINATE  CONCEPTS. —  Two  or  more  concepts  are 
said  to  l>e  co-ordinate  to  each  other,  when  they  are  included 
in  the  extent  of  the  same  concept,  hut  at  the  same  time  ex- 
clude each  other. 

Thus,  the  two  concepts  plant  and  animal  are  coordinate 
to  each  other,  being  both  species  of  the  same  genus  organic 
heing,  and  also  excluding  each  other.  The  terms  plant 
and  tree  are  not  coordinate.  They  are  both  included  in 
the  extent  of  organic  heing,  but  they  do  not  exclude  each 
other.  As  another  example  of  coordinate  terms  we  may 
take^sA,  hird,  and  mammal,  all  three  being  species  of  the 
genus  vertebrate.  The  relation  between  coordinate  terms 
is  shown  by  the  diagram,  where  A  represents  the  genus 
and  B,  C,  and  D  three  of  its  species. 


CONCEPTS.  \  3 


If  one  concept  is  subordinate  to  another,  they  must  be 
compatible  y  and  if  tivo  or  more  concepts  are  co-ordinate  to 
each  other,  they  must  be  incompatible. 

The  truth  of  this  may  be  verified  by  taking  the  terms 
parallelogram  and  quadrilateral  and  the  terms  plant  and 
animal.  Of  the  two  terms  parallelogram  and  quadrilat- 
eral, the  former  is  subordinate  to  the  latter.  But  as  all 
rectangles  are  parallelograms,  and  also  all  rectangles  are 
quadrilaterals,  we  see  that  the  two  terms  parallelogram 
and  quadrilateral  may  both  be  affirmed  of  the  same  sub- 
ject, rectangle.  Hence  parallelogram  and  quadrilateral  are 
compatible. 

The  two  terms  plant  and  animal  are  evidently  coordi- 
nate to  each  other,  both  being  species  of  the  genus  organic 
being,  and  at  the  same  time  excluding  each  other.  But 
we  cannot  find  any  subject  of  which  they  may  both  be 
affirmed.  Hence  they  are  incompatible. 


CHAPTER  III. 


JUDGMENTS. 

1.  Definition  of  Judgment. 

Judgment  is  that  act  of  thought  by  which  we  compare 
two  objects  of  thought,  asserting  whether  they  agree  or  not. 

The  product  of  this  operation  is  called  a  judgment.  A 
logical  proposition  is  a  judgment  expressed  in  words.  For 
instance,  All  horses  are  mammals. 

2.  Terms. 

Every  judgment  contains  two  ideas,  called  the  terms  of 
the  judgment.  The  term  of  which  something  is  affirmed 
or  denied  is  called  the  subject,  and  the  term  which  is  af- 
firmed or  denied  of  the  subject  is  called  the  predicate. 
The  word  that  expresses  the  connection  between  the  sub- 
ject and  the  predicate  is  called  the  copula. 

Thus,  in  the  judgment 

Man  is  mortal, 
man  is  the  subject,  mortal  the  predicate,  and  is  the  copula. 

Of  the  two  terms  of  a  judgment  the  predicate  is  usually 

(14) 


JUDGMENTS.  15 


a  concept,  and  the  subject  may  be  either  a  concept  or  a 
percept.  Thus,  in  the  proposition  Insects  are  animals, 
both  the  subject  and  the  predicate  are  concepts.  In  the 
proposition  Chicago  is  a  city,  the  subject  is  a  percept  and 
the  predicate  a  concept.  Sometimes  both  terms  may  be 
percepts,  as  in  Chicago  is  not  London. 

3.   Classification  of  Judgments. 

Judgments  are  classified  according  to  quality,  quantity, 
relation,  and  modality. 

1.  QUALITY. — According  to  quality  judgments  are  di- 
vided into  affirmative  and  negative. 

d)  An  affirmative  judgment  is  one  in  which  the  predi- 
cate is  affirmed  of  the  subject. 

For  instance, 

All  horses  are  animals. 

b)  A  negative  judgment  is  one  in  which  the  predicate  is 
denied  of  the  subject. 
For  instance, 

No  roses  are  animals. 

2.  QUANTITY.  —  According  to   quantity  judgments   are 
divided  into  universal  and  particular. 

a)  A  universal  judgment  is  one  in  which  the  predicate 
is  affirmed  or  denied  of  the  subject  in  its  whole  extent. 
For  instance, 

All  men  are  mortal. 


OUTLINES  OF  LOGIC. 


J)  A  particular  judgment  is  one  in  which  the  predicate 
is  affirmed  or  denied  of  the  subject  only  in  part  of  its  ex- 
tent. 

For  instance, 

Some  animals  are  insects. 
No  triangles  are  circles. 

A  judgment  which  has  for  its  subject  a  singular  term  is 
sometimes  called  a  singular  judgment,  as  Alexander  was  a 
conqueror.  All  singular  judgments,  however,  are  univer- 
sal, since  in  such  a  judgment  the  predicate  is  evidently 
affirmed  or  denied  of  the  whole  of  the  subject. 

A  proposition  is  said  to  be  indefinite  when  it  has  no 
mark  of  quantity  whatever,  leaving  it  ambiguous  whether 
it  is  universal  or  particular.  In  all  such  cases,  however, 
the  proper  mark  of  quantity  can  be  prefixed.  Thus,  the 
indefinite  proposition  Man  is  mortal  means  All  men  are 
mortal. 

The  combination  of  difference  in  quality  with  difference 
in  quantity  gives  rise  to  four  classes  of  judgments  : 
Universal  affirmative.  A. 
Universal  negative.  E. 
Particular  affirmative.  I. 
Particular  negative.    O. 

These  four  classes  of  judgments  are  designated  by  the 
letters  A,  E,  I,  and  O.  It  is  easy  to  remember  what 
kind  of  judgment  each  letter  represents  by  observing  that 
A  and  I  are  the  first  two  vowels  of  the  Latin  word  affirmo, 


JUDGMENTS.  17 


and  E  and  O  the  vowels  of  nego.     We  give  the  following 
examples : 

All  insects  are  animals.  A. 

No  men  are  gods.  E. 

Some  men  are  wise.     I. 

Some  men  are  not  wise.    O. 

In  passing  from  a  particular  affirmative  to  a  particular 
negative  judgment,  we  prefix  not  to  the  predicate.  When 
we  pass  from  a  universal  affirmative  to  a  universal  nega- 
tive judgment,  however,  this  is  not  sufficient.  In  that 
case  the  negative  adjective  no  must  be  prefixed  to  the 
subject.  Let  us  take  the  universal  affirmative  judgment 
All  men  are  rational.  By  prefixing  not  to  the  predicate 
we  have  All  men  are  not  rational,  which  may  be  particu- 
lar and  may  imply  that  some  men  may  be  rational.  It  is 
therefore  not  a  complete  negation  of  the  universal  affirm- 
ative judgment  All  men  are  not  rational.  Hence,  in 
order  to  express  a  complete  denial  of  the  universal  affirm- 
ative judgment  we  must  prefix  no  to  the  subject.  Thus, 
No  men  are  rational. 

3.  RELATION. — According  to  relation  judgments  are  di- 
vided into  categorical,  hypothetical,  and  disjunctive. 

a)  A  categorical  judgment  is  one  in  which  the  predicate 
is  unconditionally  affirmed  or  denied  of  the  subject. 

The  simplest  form  of  a  categorical  judgment  is, 
S  is  P. 

-2 


18  OUTLINES  OF  LOGIC. 

For  instance, 

All  trees  are  plants. 

Some  heavenly  bodies  are  not  planets. 

5)  A;  hypothetical  judgment  is  one  in  which  the  predi- 
cate is  affirmed  or  denied  of  the  subject  conditionally. 
The  simplest  form  of  a  hypothetical  judgment  is, 
If  A  is  B,  Cis  D. 

A  hypothetical  judgment  thus  consists  of  two  categor- 
ical judgments  connected  by  the  conjunction  if.  The 
first,  or  the  one  that  expresses  the  condition,  is  called  the 
antecedent^  and  the  other  the  consequent. 

For  instance, 

If  rain  does  not  come,  the  crops  will  fail. 

Here,  If  rain  does  not  come  is  the  antecedent,  and  the 
crops  will  fail  is  the  consequent. 

A  hypothetical  judgment  can  always  be  changed  to  a 
categorical  judgment  of  exactly  the  same  meaning,  having 
for  its  subject  the  antecedent  and  for  its  predicate  the 
consequent  of  the  hypothetical  judgment. 

Thus,  the  hypothetical  judgment 

If  a  triangle  is  equilateral,  it  is  equiangular 
can  be  converted  into  the  categorical  judgment 
All  equilateral  triangles  are  equiangular. 

c)  A  disjunctive  judgment  is  one  that  expresses  an 
alternative. 


JUDGMENTS.  19 


The  simplest  form  of  a  disjunctive  judgment  is 
S  is  either  P  or  not  P. 

The  disjunctive  judgment  has  instead  of  a  single  predi- 
cate two  alternatives  or  more,  of  which  one  must  be 
asserted  of  the  subject  to  the  exclusion  of  any  other  alter- 
native. For  instance, 

John  is  either  in  the  house  or  not  in  the  house. 

This  triangle  is  either  right-angled,  obtuse-angled,  or 
acute-angled. 

A  disjunctive  judgment  is  called  divisive  when  the  predi- 
cate expresses  all  the  species  of  the  subject.     For  instance, 
Organic  beings  are  divided  into  animals  and  plants. 
Triangles  are  divided  into  right-angled  and  oblique- 
angled. 

The  divisive  judgment  is  disjunctive  only  in  form,  but 
categorical  in  sense.     It  is,  in  reality,  composed  of  two  or 
more  particular  judgments.     Thus,  the  judgment  Triangles 
are  divided  into  right-angled  and  oblique-angled  is  com- 
posed of  the  two  particular  judgments 
Some  triangles  are  right-angled. 
Some  triangles  are  oblique-angled. 

4.  MODALITY. — According  to  modality^  or  the  degree  of 
certainty,  judgments  are  divided  into  apodictic,  problematic, 
and  assert ory. 

a)  An  apodictic  judgment  is  one  which  expresses  the 


20  OUTLINES  OF  LOGIC. 

combination  between  the  subject  and  the  predicate  as  a 
necessity. 

8  must  be  P. 
For  instance, 

An  equilateral  triangle  must  be  equiangular. 

b)  A  problematic  judgment  is  one  ivhich  expresses  the 
combination  between  the  subject  and  the  predicate  as  a  pos- 
sibility. 

8  may  be  P. 
For  instance, 

Mars  may  be  inhabited. 

c]  An  assertory  judgment  is  one  which  expresses  the 
combination  between  the  subject  and  the  predicate  as  a  fact 
to  be  taken  for  granted. 

8  is  P. 
For  instance, 

This  dog  is  mad. 

4.   Distribution  of  Terms. 

A  term  is  said  to  be  distributed  when  it  is  taken  uni- 
versally or  in  its  whole  extent. 

For  instance,  in  the  judgment  All  animals  are  organic 
beings,  the  term  animal  is  taken  universally  or  in  its  whole 
extent,  and  is  therefore  distributed. 

1.  With  regard  to  the  subject  we  have  the  following 
rules : 

a)     In         All  8  are  P     (A) 
and       No  8  are  P     (E) 
the  subject  is  distributed,  both  judgments  being  universal. 


JUDGMENTS.  21 


b)     In         Some  S  are  P    (I) 

and       Some  S  are  not  P    (0) 

the  subject  is  not  distributed,  both  judgments  being  partic- 
ular. 

2.  With  regard  to  the  predicate  we  have  the  following 
rules : 

a)  In         All  S  are  P    (A) 

the  predicate  is  not  distributed.  It  is  evident  that  the 
whole  of  P  is  not  considered,  as  P  may  contain  many 
other  things  besides  /S. 

b)  In         Same  8  are  P    (I) 

the  predicate  is  not  distributed,  as  is  shown  by  the  same 
reasoning  as  for  A. 

c)  In         No  S  are  P    (E) 

the  predicate  is  distributed.  In  order  to  assert  that  no 
part  of  8  belongs  to  any  part  of  P,  it  is  evident  that  the 
whole  of  P  must  be  considered. 

d)  In         Some  S  are  not  P    (O) 

the  predicate  is  distributed.  The  same  reasoning  applies 
here  as  for  E.  We  must  consider  the  whole  of  P  in  or- 
der to  assert  that  no  part  of  it  belongs  to  some  P  in  ques- 
tion. 

REMARKS. 

1.  A  distributes  the  predicate  in  case  the  subject  and 
the  predicate  are  co-extensive,  i.  e.,  have  exactly  the  same 
extent. 

For  instance, 

All  equilateral  triangles  are  equiangular. 


22 


OU7  LINES  OF  LOGIC. 


2.  I  distributes  the  predicate  in  case  the  subject  is  the 

genus  and  the  predicate  one  of  its  species.     For  instance, 
Some  animals  are  vertebrates. 

For  the  distribution  of  terms  in  the  four  categorical  judg- 
ments we  have  then  the  following  rules : 

1.  Universal  judgments  distribute  the  subject ;  par- 

ticular judgments  do  not. 

2.  Negative  judgments  distribute  the  predicate ;  af- 

firmative judgments  do  not. 
These  rules  may  be  stated  by  the  following  schedule  : 


SUBJECT. 


A.  Distributed. 
E.  Distributed. 
I.    Tin  distr  ibuted. 
0.   Undistributed. 


PREDICATE. 

Undistributed. 
Distributed. 
Undistributed. 
Distributed. 


In  the  diagrams  given  below  the  distribution  of  the  sub- 
ject and  the  predicate  in  the  four  categorical  judgments  is 
shown.     S  represents  the  subject  and  P  the  predicate. 
A.  E. 


JUDGMENTS. 


L 


5.  Immediate  Inference. 

Immediate  inference  is  that  act  of  thought  ~by  which  we 
transform  one  judgment  into  another  and  from  the  validity 
or  invalidity  of  one  infer  the  validity  or  invalidity  of  the 
other. 

We  will  treat  immediate  inference  under  the  following 
heads : 

1.  Synonymous  Judgments. 

2.  Subalterns. 

3.  Opposition. 

4.  Conversion. 

1.  SYNONYMOUS  JUDGMENTS. — Two  judgments  are  synon- 
ymous when  they  express  the  same  fact  in  different  words. 

The  wording  of  a  proposition  may  evidently  be  changed 
in  many  different  ways  so  as  to  give  a  new  proposition, 
differing  only  in  form  but  not  in  sense  from  the  given 


24  OUTLINES  OF  LOGIC. 

one.  We  may,  for  instance,  substitute  for  either  the  sub- 
ject or  the  predicate  equivalent  terms;  or  change  from  a 
categorical  to  a  hypothetical  proposition,  and  conversely ; 
or  instead  of  affirming  one  thing,  deny  its  opposite.  Evi- 
dently both  are  true  or  both  false  at  the  same  time. 
For  instance : 

(True)    This  is  a  triangle. 

(True)    This  is  a  figure  having  three  sides. 

(False)  All  triangles  are  equilateral. 
(False)  No  triangles  are  not  equilateral. 

(True)    Damp  gunpowder  will  not  explode. 
(True)    If  gunpowder  is  damp,  it  will  not  explode. 

2.  SUBALTERNS. — Two  judgments  are  said  to  ~be  subalterns 
when  they  have  the  same  subject,  the  same  predicate,  and 
the  same  quality,  but  one  is  universal  and  the  other  partic- 
ular. 

Thus,  A  and  /  are  a  pair  of  subalterns ;  also  E  and  0. 
I  and  0  are  called  the  subalternates  of  A  and  E  respect- 
ively, each  of  which  is  a  subalternans. 

From  the  truth  of  the  universal  we  infer  the  truth  of 
the  particular,  and  from  the  falsity  of  the  particular  we 
infer  the  falsity  of  the  universal.  But  the  truth  of  the 
particular  does  not  always  include  the  truth  of  the  uni- 
versal; nor  does  the  falsity  of  the  universal  always  include 
the  falsity  of  the  particular. 


JUDGMENTS.  25 


For  instance, 

(True)   All  men  are  mortal.     (A) 
(True)   Some  men  are  mortal.     (/) 

(True)   No  animal  is  rational.     (E) 
(True)   Some  animals  are  not  rational.     (0) 

(False)  Some  plants  are  animals.     (/) 
(False)  All  plants  are  animals.     (A) 

(False)  Some  triangles  are  not  figures.     ((9) 
(False)  No  triangles  are  figures.     (E) 

But  the  truth  of  the  particular  judgment 

Some  animals  are  insects  (/) 
does  not  involve  the  truth  of  the  universal 

All  animals  are  insects.  (A) 
Nor  can  we  from  the  falsity  of  the  universal  judgment 

No  figures  are  triangles  (E) 
infer  the  falsity  of  the  particular 

Some  figures  are  not  triangles.  (0) 
Hence  we  conclude  from  the  truth  of  A  and  E  to  the 
truth  of  /  and  0  respectively,  and  from  the  falsity  of  1 
and  O  to  the  falsity  of  A  and  E  respectively ;  but  not 
from  the  falsity  of  A  and  E  to  the  falsity  of  /  and  O  re- 
spectively, nor  from  the  truth  of  /  and  0  to  the  truth  of 
A  and  E  respectively. 

3.  OPPOSITION.  —  Opposition  takes  place  between  two 
judgments  when  they  have  the  same  subject  and  the  same 
predicate,  but  opposite  quality. 


26  OUTLINES  OF  LOGIC. 

There  are  three  kinds  of  opposition  depending  on  the 
quantity  of  the  judgments,  viz.,  contrary,  contradictory, 
and  subcontrary. 

a)  Contrary. — If  both  judgments  are  universal,  the  op- 
position is  said  to  be  contrary,  or  the  judgments  are  con- 
traries each  of  the  other. 

Two  contrary  judgments  cannot  both  be  true,  but,  they 
may  both  be  false. 

Hence  the  truth  of  one  involves  the  falsity  of  the  other, 
but  the  falsity  of  one  does  not  necessarily  involve  the  truth 
of  the  other. 

For  instance, 

(True)  All  trees  are  plants.   (A) 
(False)  No  trees  are  plants.   (E) 
(True)  No  animals  are  plants.  (E) 
(False)  All  animals  are  plants.  (A) 
But  from  the  falsity  of 

All  animals  are  insects  (A) 
we  cannot  infer  the  truth  of 

No  animals  are  insects.   (.E) 
Nor  can  we  from  the  falsity  of 

No  animals  are  birds  (E) 
infer  the  truth  of 

All  animals  are  birds.   (A) 

Hence  we  conclude  from  the  truth  of  A  to  the  falsity  of 
E  and  from  the  truth  of  E  to  the  falsity  of  A,  but  not 


JUDGMENTS.  27 


from  the  falsity  of  A  to  the  truth  of  E,  nor  from  the 
falsity  of  E  to  the  truth  of  A. 

b)   Contradictory. —  If  one  judgment  is  universal  and 
the  other  particular,  the  opposition  is  said  to  be  contradic- 
tory, or  the  judgments  are  contradictories  each  of  the  other. 
Of  two  contradictory  judgments  one  must  be  true  and 
the  other  false. 

Hence  the  truth  of  one  involves  the  falsity  of  the  other, 
and  the  falsity  of  one  involves  the  truth  of  the  other. 
For  instance, 

(True)  All  plants  are  organic  beings.     (A) 
(False)  Some  plants  are  not  organic  beings.     (O) 
(True)  No  triangles  are  squares.     (E) 
(False)  Some  triangles  are  squares.     (/) 
(True)  Some  animals  are  not  birds.     (O) 
(False)  All  animals  are  birds.     (A) 
(True)  Some  plants  are  water-plants.     (/) 
(False)  No  plants  are  water-plants.     (E) 

Hence  we  conclude  from  the  truth  or  falsity  of  A,  E,  1, 
and  0  to  the  falsity  or  truth  of  <9,  /,  E,  and  A  respect- 
ively. 

(?)  Subcontrary. —  If  both  judgments  are  particular,  the 
opposition  is  said  to  be  subcontrary,  or  the  judgments  are 
subcontraries  each  of  the  other. 

Two  subcontrary  judgments  may  both  be  true,  but  they 
cannot  both  be  false. 


28  OUTLTNES  OF  LOGIC. 

» 

Hence  from  the  falsity  of  one  we  infer  the  truth  of  the 
other,  hut  the  truth  of  one  does  not  necessarily  involve  the 
falsity  of  the  other. 

For  instance, 

(False)  Some  triangles  are  not  figures.     (O) 
( True)  Some  triangles  are  figures.     (/) 

(False)  Some  plants  are  animals.     (7) 
(True)  Some  plants  are  not  animals.     (0) 

But  from  the  truth  of 

Some  heavenly  bodies  are  planets     (/) 
we  cannot  infer  the  falsity  of 

Some  heavenly  bodies  are  not  planets.     (0) 
Nor  can  we  from  the  truth  of 

Some  animals  are  not  fishes     (0) 
infer  the  falsity  of 

Some  animals  are  fishes.     (7) 

Hence  we  conclude  from  the  falsity  of  0  and  /  to  the 
truth  of  7  and  0  respectively,  but  not  from  the  truth  of  O 
and  /  to  the  falsity  of  /  and  O  respectively. 


JUDGMENTS. 


29 


The  relations  between  the  four  judgments  A,  E,  1,  and 
O  are  shown  by  the  following  schedule: 


4.  CONVERSION. — A  judgment  is  said  to  undergo  conver- 
sion or  to  be  converted  when  its  subject  and  predicate  are 
interchanged. 

If  the  given  judgment  is  true,  the  new  judgment  must 
also  be  true. 


30 


OUTLINES  OF  LOGIC. 


There  are  three  kinds  of  conversion  :  simple  conversion, 
conversion  by  limitation,  and  conversion  by  contraposition. 

a)  Simple  conversion. — A  judgment  is  simply  converted 
when  its  subject  and  predicate  are  interchanged,  the  qual- 
ity and  quantity  remaining  the  same. 

For  instance, 


No  metals  are  compounds.  (E) 
No  compounds  are  metals.  (E) 


Some  flowers  are  yellow.   (/) 
Some  yellow  things  are  flowers.  (/) 


But  from  the  judgment 


All  metals  are  elements  (A) 
we  cannot  infer  that 
All  elements  are  metals. 


Nor  can  we  pass  from 

Some  plants  are  not  water-plants  (O) 

to 

Some  water-plants  are  not  plants. 

Hence  only  universal  negative  and  particular  affirma- 
tive judgments  can  be  simply  converted. 


JUDGMENTS. 


fy  Conversion  by  limitation. — A  judgment  is  said  to  be 
converted  by  limitation  when  its  subject  and  predicate  are 
interchanged,  the  quality  remaining  the  same,  but  the 
quantity  being  changed. 

For  instance, 


All  men  are  mortal.     (A) 

Some  mortal  beings  are  men.     (/) 


But  from  the  judgment 


Some  animals  are  not  insects  (O) 

we  cannot  pass  to 

No  insects  are  animals. 


Hence  all  universal  affirmative  judgments  can  be  con- 
verted by  limitation.  To  particular  negative  judgments 
neither  simple  conversion  nor  conversion  by  limitation 
can  be  applied. 

There  are,  however,  some  universal  affirmative  judg- 
ments that  can  be  simply  converted  ;  namely,  all  those  in 
which  the  subject  and  the  predicate  are  co-extensive.  To 
that  class  belong  all  logical  definitions. 

For  instance, 

A  quadrilateral  is  a  figure  having  four  sides. 
All  figures  having  four  sides  are  quadrilaterals. 


OUTLINES  OF  LOGIC. 


All  equilateral  triangles  are  equiangular. 
All  equiangular  triangles  are  equilateral. 

c)  Conversion  by  contraposition. — We  are  said  to  con- 
vert a  judgment  by  contraposition  when  we  first  change 
the  quality  and  for  the  predicate  substitute  its  contradic- 
tory and  then  apply  simple  conversion. 

By  the  first  process  we  pass  from  the  affirmation  of  one 
thing  to  the  denial  of  its  opposite.  For  instance, 


All  metals  are  elements  (A) 
No  metals  are  not-elements  (E) 


and  then  by  simple  conversion 

No  not-elements  are  metals  (E) 


Some  animals  are  not  insects  (0) 
Some  animals  are  not-insects  (/) 


and  by  simple  conversion 

Some  not-insects  are  animals  (/). 

In  the  particular  negative  judgment  we  thus  simply 
transfer  the  negative  particle  from  the  copula  to  the  pred- 
icate and  then  apply  simple  conversion. 


JUDGMENTS.  33 


Hence  all  universal  affirmative  and  particular  negative 
judgments  can  be  converted  by  contraposition. 

A  similar  process  may  be  applied  to  the  universal  neg- 
ative judgment,  though  in  that  case  we  can  only  convert 
by  limitation.  For  instance, 


No  fishes  are  birds.  (E) 
All  fishes  are  not-birds.  (A) 


and  by  conversion  by  limitation 

Some  not-birds  are  fishes.  (/) 

For  conversion  we  have  then  the  following  rules : 

I.  Only  universal  negative  and  particular  affirmative 
judgments  can  be  simply  converted. 

II.  All  universal  affirmative  judgments  can  be  converted 
l>y  limitation. 

III.  Particular  negative  judgments  can  only  be  converted 
by  contraposition. 

6.  Simple  and  Complex  Propositions. 

A  simple  proposition  is  one  that  has  only  one  subject 
and  one  predicate.     For  instance, 

Gold  is  a  metal. 
A  complex  proposition  is  one  that  has  more  than  one 

—3 


34  OUTLINES  OF  LOGIC. 

subject,  or  more  than   one   predicate,    or   both.     For  in- 
stance, 

Birds  and  fishes  are  animals. 

In  this  example  there  are  evidently  two  categorical  prop- 
ositions combined  in  one,  viz., 

Birds  are  animals 
and  Fishes  are  animals. 

The  only  complex  propositions  with  which  logic  is  di- 
rectly concerned  are  the  hypothetical  and  the  disjunctive 
propositions,  which  have  already  been  described. 


CHAPTER  IV. 


SYLLOGISMS. 

1.  Definition  of  Syllogism. 

Syllogism  is  the  process  by  which  two  objects  of  thought 
are  compared  through  their  relation  to  a  third. 

Every  syllogism  contains  three  terms,  the  major  term, 
the  middle  term,  and  the  minor  term.  The  relation  be- 
tween the  three  terms  is  expressed  by  three  judgments,  of 
which  two  are  called  the  premises  and  the  third  the  conclu- 
sion. In  one  premise  the  middle  term  is  compared  with 
the  major  term,  in  the  other  premise  it  is  compared  with 
the  minor  term,  and  in  the  conclusion  the  major  and 
minor  terms  are  compared.  The  premise  containing  the 
major  term  is  called  the  major  premise,  and  the  premise 
containing  the  minor  term  is  called  the  minor  premise. 
The  middle  term,  being  only  the  medium  of  comparison 
between  the  two  other  terms,  occurs  only  in  the  premises, 
but  not  in  the  conclusion.  The  minor  term  is  always  the 
subject  of  the  conclusion,  and  the  major  term  is  always  the 
predicate  of  the  conclusion. 

(35) 


36  OUTLINES  OF  LOGIC. 

The  minor  and  major  terms  are  so  called  because  the 
major  term  has  usually  greater  extent  than  the  minor 
term.  The  three  terms  of  a  syllogism  are  usually  repre- 
sented by  the  letters  P,  M,  and  S.  P  designates  the 
major  term,  being  the  predicate  of  the  conclusion ;  M  de- 
notes the  middle  term ;  and  8  denotes  the  minor  term, 
being  the  subject  of  the  conclusion. 

The  three  judgments  of  a  syllogism  are  usually  arranged 
in  the  following  order : 

Major  premise.  All  men  are  rational. 
Minor  premise.  All  Americans  are  men. 
Conclusion.  All  Americans  are  rational. 

In  the  example  given  above,  men  is  the  middle  term, 
rational  the  major  term,  and  Americans  the  minor  term. 

The  syllogism  may  also  be  defined  as  the  act  of  thought 
by  which  from  two  given  judgments,  called  the  premises, 
we  draw  or  infer  a  third  judgment,  called  the  conclusion. 
Syllogism  is  also  called  mediate  inference,  and  differs  from 
immediate  inference,  described  in  the  preceding  chapter, 
mediate  inference  being  made  through  a  medium  or  a 
middle  term. 

2.  Classification  of  Syllogisms. 

Syllogisms  are  divided  into  categorical,  hypothetical,  and 
disjunctive. 

1.  A  categorical  syllogism  is  a  syllogism  having  for  its 
major  premise  a  categorical  judgment. 


SYLLOGISMS.  37 


2.  A  hypothetical  syllogism  is  a  syllogism  having  for 
its  major  premise  a  hypothetical  judgment. 

3.  A  disjunctive  syllogism  is  a  syllogism  having  for  its 
major  premise  a  disjunctive  judgment. 

Examples: 

r     M  is  p. 

Categorical.       \        S  is  M. 
[  /.  S  is  P. 

f       If  A  is  B,  C  is  D. 
Hypothetical.     \        A  is  B. 
[  .-.  C  is  D. 

A  is  either  B  or  not-B. 
Disjunctive.       {        A  is  B. 

.*.  A  is  not  not-B. 

3.  Categorical  Syllogisms. 

1.  DEFINITION. — A  categorical  syllogism  is  a  syllogism 
having  for  its  major  premise  a  categorical  judgment. 

The  minor  premise  and  the  conclusion  are  also  categor- 
ical judgments. 

2.  RULES. — A  general  rule  for  the  syllogism  is  an  axiom 
known  as  the  dictum  de  omni  et  nullo  of  Aristotle.     This 
axiom  may  be  stated  thus : 

Whatever  is  affirmed  or  denied  of  a  whole  class  may  also 
l)e  affirmed  or  denied  of  any  individual  contained  in  that 


38  OUTLINES  OF  LOGtC. 

The  special  rules  of  the  categorical  syllogism  are : 

I.  The  syllogism  must  contain  three  and  only  three  terms. 

II.  The  syllogism  must   contain  three  and  only  three 
judgments. 

III.  The  middle  term  must  he  distributed  at  least  in  one 
of  the  premises. 

IY.  In  order  that  a  term  may  he  distributed  in  the  con- 
clusion it  must  be  distributed  in  one  of  the  premises. 

V.  From  two  negative  premises  no  conclusion  can  he 
drawn. 

YI.  From  two  particular  premises  no  conclusion  can  he 
drawn. 

YII.  If  one  premise  is  negative  the  conclusion  will  he 
negative. 

VIII.  If  one  premise  is  particular  the  conclusion  will 
he  particular. 

3.  EXPLANATION  OF  THE  RULES. — The  first  and  second 
rules  need  no  further  explanation. 

3d  rule.  If  the  middle  term  were  not  distributed  in  at 
least  one  of  the  premises,  it  might  happen  that  the  minor 
and  major  terms  are  compared  with  different  parts  of  the 
middle  term,   and  therefore  the  middle  term   would    no 
longer  be  a  medium  of  comparison.     For  instance, 
All  P  are  M 
All  S  are  M. 

Here  the  middle  term  is  not  distributed.     F  is  one  part 


SYLLOGISMS.  39 


of  M  and  S  is  another  part  of  M,  and  these  parts  may  or 
may  not  coincide.  No  relation  can  be  established  between 
S  and  P,  as  S  may  fall  wholly  without,  or  wholly  within, 
or  partly  without  and  partly  within  P,  as  is  seen  in  the  di- 
agram. 


rule.  If  either  the  major  or  minor  term  is  not  dis- 
tributed in  the  premise  where  it  occurs,  it  must  not  be 
distributed  in  the  conclusion.     It  is  evident  that  we  are 
only  enabled  to  infer  something  about  that  part  of  either 
the  major  or  minor  term  which  has  been  compared  with 
the  middle  term  in  the  premise.     In  the  syllogism 
All  insects  are  animals 
No  dogs  are  insects 
.•.  No  dogs  are  animals 

the  major  term  animals  is  not  distributed  in  the  major 
premise,  but  is  distributed  in  the  conclusion.  This  argu- 
ment is  consequently  fallacious.  This  fallacy  is  called  an 
illicit  process  of  the  major  term. 


OUTLINES  OF  LOGIC. 


Again,  in  the  example 

All  flies  are  insects 

All  flies  are  animals 

.'.  All  animals  are  insects 

the  minor  term  animals  is  distributed  in  the  conclusion, 
but  not  in  the  minor  premise.  Hence  the  argument  is 
false.  This  kind  of  fallacy  is  called  an  illicit  process  of 
the  minor  term. 

5th  rule.  If  both  premises  are  negative,  no  conclusion 
can  be  drawn,  because  the  middle  is  no  longer  a  medium 
of  comparison  between  the  minor  and  major  terms.  For 
instance,  from  the  premises 

No  M  are  P 

No  S  are  M 

no  conclusion  can  be  drawn  as  regards  the  relation  be- 
tween S  and  P,  as  S  may  fall  wholly  within,  or  wholly 
without,  or  partly  within  and  partly  without  P,  as  is  seen 
in  the  diagram. 


SYLLOGISMS.  41 


6th  rule.  If  both  premises  are  particular,  no  conclusion 
can  be  drawn,  because  no  relation  can  be  established  be- 
tween two  terms  that  are  only  partly  connected  with  a 
third.  From  the  premises 

Some  M  are  P 

Some  S  are  M 

no  conclusion  can  be  drawn ;  for,  as  is  shown  by  the  dia- 
gram, S  may  fall  wholly  without,  or  wholly  within,  or 
partly  without  and  partly  within  P. 


7th  rule.  If  the  minor  premise,  for  instance,  be  negative, 
thus  expressing  a  disagreement  between  the  minor  and 
middle  terms,  and  the  major  premise  affirmative,  express- 
ing an  agreement  between  the  major  and  middle  terms, 
the  conclusion  must  necessarily  express  a  disagreement 
between  the  minor  and  major  terms,  i.  e.,  the  conclusion 
must  be  negative.  And  in  the  same  way  if  the  major 
premise  is  negative  and  the  minor  premise  affirmative. 

8th  rule.  This  rule  is  in  fact  a  corollary  of  the  third  and 
fourth  rules. 


OUTLINES  OF  LOGIC. 


4.  FIGURES.— The  three  terms  of  a  syllogism  may  be 
arranged  in  different  ways.  It  is  evident  that  the  middle 
term  can  have  only  four  different  positions,  and  hence 
there  are  four  different  ways,  or,  as  they  are  called,  figures 
in  which  the  terms  of  a  syllogism  may  be  arranged. 
These  four  figures  of  the  syllogism  are  shown  in  the  fol- 
lowing scheme : 


Mis  P. 
SisM. 

PisM. 

S  is  M. 

Mis  P. 

Mis  S. 

PisM. 

MisS. 

Sis  P. 

S  is  P. 

S  is  P. 

Sis  P. 

In  the  first  figure  the  middle  term  is  the  subject  of  the 
major  premise  and  the  predicate  of  the  minor  premise. 

In  the  second  figure  the  middle  term  is  the  predicate  of 
both  the  major  and  the  minor  premises. 

In  the  third  figure  the  middle  term  is  the  subject  of 
both  the  major  and  the  minor  premises. 

In  the  fourth  figure  the  middle  term  is  the  predicate  of 
the  major  premise  and  the  subject  of  the  minor  premise. 

The  first  three  figures  were  proposed  by  Aristotle,  and 
hence  they  are  usually  called  the  Aristotelian  figures.  The 
fourth  figure,  proposed  by  Galen,  is  really  an  inversion  of 
the  first  figure  and  is  comparatively  useless,  because  the 
same  conclusions  can  be  obtained  more  naturally  by  using 
the  first  figure. 


SYLLOGISMS.  43 


5.  MOODS. — As  every  syllogism  must  contain  two  prem- 
ises and  each  premise  may  be  either  universal  affirmative, 
universal  negative,  particular  affirmative,  or  particular 
negative,  there  would  be  in  each  figure  sixteen  different 
forms  of  the  syllogisms,  or,  as  they  are  called,  m,oods,  de- 
pending on  the  quality  and  quantity  of  the  premises.  But 
the  number  of  moods  in  each  figure  is  limited  by  the  rules 
of  the  syllogism  mentioned  above,  and  thus  omitting  all 
moods  which  violate  these  rules  and  all  moods  which  are 
useless,  being  included  in  other  moods,  there  will  remain 
only  nineteen.  As  an  artificial  aid  in  memorizing  these 
nineteen  possible  moods  the  following  mnemonic  verses 
have  been  invented : 

Fig.  1.  Barbara,  Celarent,  Darii,  Ferioque,  prioris ; 

Fig.  2.    Cesare,  Carnestres,  Festino,  Baroko,  secundse ; 

Fig.  3.  Tertia,  Darapti,  Disamis,  Datisi,  Felapton,  Bo- 
kardo,  Ferison  habet ;  quarta  insuper  addit, 

Fig.  4.  Bramantip,  Camenes,  Dimaris,  Fesapo,  Fresison. 

Each  of  the  italicized  names  represents  a  mood,  the 
vowels  of  each  name  standing  for1  the  three  judgments  of 
the  syllogism.  Thus  for  instance  Cesare  signifies  the 
mood  of  the  second  figure,  which  has  E  for  the  major 
premise,  A  for  the  minor,  and  E  for  the  conclusion. 

First  Figure. 
This  is  the  only  figure  in  which  the  conclusion  can  be 


44: 


OUTLINES  OF  LOGIC. 


universal  affirmative.     With  regard  to  the  premises  the 
following  rules  will  be  observed : 

a)  The  major  premise  must  be  universal. 

b)  The  minor  premise  must  be  affirmative. 

These  special  rules  can  easily  be  deduced  from  the  gen- 
eral rules  of  the  syllogism. 

The  four  valid  moods  in  this  figure  are : 

Barbara,  Celarent,  Darii,  and  Ferio. 


Barbara. 


EXAMPLES. 


All  men  are  mortal. 
All  Americans  are  men. 
/.  All  Americans  are  mortal. 


Celarent. 


Darii. 


No  quadrilaterals  are  cir- 
cles. 

A 1 1  parallelograms  are 
quadrilaterals. 

No  parallelograms  are  cir- 
cles. 


All  mammals  have  red  blood. 
Some  animals  are  mammals. 
Some  animals  have  red  blood. 


SYLLOGISMS. 


45 


Ferio. 


No  insects  are  warm-blooded. 
Some  animals  are  insects. 
.*.  Some  animals  are  not  warm- 
blooded. 


Second  figure. 

In  this  figure  the  conclusion  is  always  negative.     For 
the  premises  we  have  the  following  special  rules: 
a)  The  major  premise  must  be  universal, 
ft)   One  of  the  premises  must  ~be  negative. 
The  four  valid  moods  of  this  figure  are : 

Cesar e,  Camestres,  Festino,  and  Baroko. 


Cesare. 


EXAMPLES. 


Cam.estres. 


No  trapezoid  is  equilat- 
eral. 

All  squares  are  equilat- 
eral. 

.'.  No    squares    are    trape- 
zoids. 


All  men  are  rational. 
No  apes  are  rational. 
.*.  No  apes  are  men. 


OUTLINES  OF  LOGIC. 


Festino. 


Baroko. 


No  planets  are  self-luminous. 
Some    heavenly    bodies    are 

self-luminous. 
/.  Some  heavenly  bodies  are  not 

planets. 


All  horses  are  mammals. 
Some  animals  are  not  mammals. 
,*.  Some  animals  are  not  horses. 


Third  Figure. 

In  this  figure  the  conclusion  is  always  particular.     For 
the  premises  we  have  the  following  rule : 

The  minor  premise  must  be  affirmative. 

Six  moods  are  possible,  viz. : 
Darapti,  Disamis,  Datisi,  Felapton,  Bokardo,  and  Ferison. 


Darapti. 


EXAMPLES. 


All  whales  are  mammals. 
All  whales  live  in  water. 
/.  Some  animals  living  in  wa- 
ter are  mammals. 


SYLLOGISMS. 


Disamis. 


Datisi. 


Felapton. 


Bokardo. 


Some  parallelograms  are  rectangles. 
All  parallelograms  are  quadrilaterals. 
Some  quadrilaterals  are  rectangles. 


All  parallelograms  are  quadri- 
laterals. 

Some  parallelograms  are  equi- 
lateral. 


,\  Some    equilateral    figures 
quadrilaterals. 


are 


No  man  is  omniscient. 
All  men  are  rational. 
Some  rational  beings  are  not  omnis- 
cient. 


Some  plants  are  not  trees. 
All  plants  are  living  beings. 
Some  living  beings  are  not  trees. 


OUTLINES  OF  LOGIC. 


Ferison. 


No  animals  are  plants. 
Some  animals  live  in  water. 
.'.  Some  organisms  living  in  water  are 
not  plants. 


Fourth  Figure. 

In  this  figure  there  are  five  valid  moods,  viz. : 
Bramantip,  Camenes,  Dimaris,  Fesapo,  and  Fresison. 


Bramantip. 


Camenes. 


EXAMPLES. 

All  fishes  breathe  by  gills. 

All  animals  breathing  by  gills  are 

cold-blooded. 
•.  Some    cold  -  blooded     animals    are 

fishes. 


All  men  are  mortal. 
No   mortal   being  is  om- 
niscient. 

.*.  No  omniscient  being  is  a 
man. 


SYLLOGISMS. 


Dimaris. 


Fesapo. 


Fresison. 


Some  taxes  are  oppressive. 

All  oppressive  things  should  be 

repealed. 

.«.  Some  things  which  should  be  re- 
pealed are  taxes. 


No  immoral  acts  are  proper  amuse- 
ments. 

All  proper  amusements  are  de- 
signed to  give  pleasure. 

Some  things  designed  to  give  pleas- 
ure are  not  immoral  acts. 


No  birds  have  gills. 

Some   animals   having   gills 

are  vertebrates. 
/.  Some    vertebrates    are    not 

birds. 


4.  Hypothetical  Syllogisms. 

1.  DEFINITION. — A  hypothetical  syllogism  is  a  syllogism 
which  has  for  its  major  premise  a  hypothetical  judgment. 

The  minor  premise  and  the  conclusion  are  usually  cate- 
gorical judgments.  If  all  three  judgments  are  hypothet- 


50  OUTLINES  OF  LOGIC. 

ical,  the  syllogism  follows  the  same  rules  as  the  categorical 
syllogism,  to  which  it  can  easily  be  reduced. 

For  instance, 

If  a  man  violates  the  laws,  he  ought  to  be  punished. 
If  a  man  commits  murder,  he  violates  the  laws. 
.*.  If  a  man  commits  murder,  he  ought  to  be  punished. 

This  can  easily  be  put  in  the  form  of  a  categorical  syl- 
logism as  follows : 

A  man  that  violates  the  laws  ought  to  be  punished. 
A  murderer  violates  the  laws. 
.*.  A  murderer  ought  to  be  punished. 

In  the  following  we  will  therefore  only  consider  Irypo- 
thetical  syllogisms  in  which  the  minor  premise  and  the 
conclusion  are  categorical  judgments. 

2.  MOODS. — Hypothetical  syllogisms  are  divided  into 
constructive  and  destructive^  according  as  the  minor  prem- 
ise is  affirmative  or  negative.  The  first  form  is  also  called 
the  modus  ponens,  or  the  mood  that  affirms,  and  the  sec- 
ond the  modus  tollens,  or  the  mood  that  denies. 

a)  Modus  ponens. —  For  this  mood  we  have  the  follow- 
ing rule  : 

If  the  antecedent  be  affirmed,  the  consequent  must  be 
affirmed.  The  minor  premise  affirms  the  antecedent  and 
the  conclusion  affirms  the  consequent. 


SYLLOGISMS.  51 


The  general  form  of  a  constructive  hypothetical  syl- 
logism is 

If  A  is  B,  Cis  D. 
A  is  B. 
/.  C  is  D. 

For  instance, 

If  a  triangle  is  equilateral,  it  is  equiangular. 
This  triangle  is  equilateral. 
,'.  This  triangle  is  equiangular. 

If  he  has  a  fever,  he  is  sick. 
He  has  a  fever. 
.*.  He  is  sick. 

5)  Modus  tollens. —  For  this  mood  we  have  the  follow- 
ing rule : 

If  the  consequent  le  denied,  the  antecedent  must  be  de- 
nied. The  'minor  premise  denies  the  consequent,  and  the 
conclusion  denies  the  antecedent. 

The  general  form  of  a  destructive  hypothetical  syl- 
logism is 

If  A  is  B,  Cis  D. 

C  is  not  D. 
/.  A  is  not  B. 
For  instance, 

If  a  triangle  is  equilateral,  it  is  equiangular. 
This  triangle  is  not  equiangular. 
/.  This  triangle  is  not  equilateral. 


52  OUTLINES  OF  LOGIC. 

If  a  man  is  a  murderer,  he  ought  to  be  punished. 
This  man  ought  not  to  be  punished. 
/.  This  man  is  not  a  murderer. 

3.  FALLACIES. — If  the  minor  premise  either  affirms  the 
consequent  or  denies  the  antecedent,  a  fallacy  of  argument 
arises.  If  we  affirm  the  consequent,  we  may  not  therefore 
affirm  the  antecedent,  because  the  consequent  might  follow 
from  some  other  antecedent  as  well  as  from  the  one  given  ; 
or,  as  we  might  express  it,  a  given  effect  may  be  produced 
by  several  different  causes.  For  the  same  reason  it  is  evi- 
dent that  we  cannot  pass  from  the  denial  of  the  antecedent 
to  the  denial  of  the  consequent.  Thus  the  argument, 

If  he  has  a  fever,  he  is  sick. 

He  is  sick. 
/.  He  has  a  fever. 

is  fallacious.  If  a  person  is  sick,  it  does  not  necessarily 
follow  that  he  has  a  fever.  He  may  be  sick  from  some 
other  cause.  For  the  same  reason  the  argument, 

If  he  has  a  fever  he  is  sick. 

He  has  not  a  fever. 
/.  He  is  not  sick, 
is  fallacious. 

There  is  one  exception  to  this  rule,  and  that  is  in  case 
the  given  condition  is  the  only  condition  of  the  consequent. 
In  such  a  case  we  may  pass  from  the  affirmation  of  the 
consequent  to  the  affirmation  of  the  antecedent,  or  from 


SYLLOGISMS.  53 


the  denial  of  the  antecedent  to  the  denial  of  the  conse- 
quent.    For  instance, 

If  a  triangle  is  equilateral,  it  is  equiangular. 

This  triangle  is  equiangular. 
/.  This  triangle  is  equilateral. 

If  a  triangle  is  equilateral,  it  is  equiangular. 
This  triangle  is  not  equilateral. 
/.  This  triangle  is  not  equiangular. 

In  the  above  examples  the  two  terms  equilateral  tri- 
angle and  equiangular  triangle  are  evidently  co-extensive. 

4.   REDUCTION   OF   HYPOTHETICAL  TO   CATEGORICAL   SYL- 
LOGISMS.—  As  we  have  already  seen,  every  hypothetical 
judgment  can   be  converted   into   a   universal   affirmative 
judgment.     Hence   every   hypothetical   syllogism   can   be 
reduced  to  the  categorical  form  and  will  consequently  fol- 
low the  rules  laid   down   for  the  categorical   syllogisms. 
In  order  to  illustrate  this  we  take  the  following  example : 
If  an  animal  is  a  mammal,  it  has  red  blood. 
All  horses  are  mammals. 
/.  All  animals  have  red  blood. 

By  changing  the  major  premise  into  a  categorical  judg- 
ment we  obtain  a  categorical  syllogism  in  the  mood 

bara. 

All  mammals  have  red  blood. 

All  horses  are  mammals. 
/.  All  horses  have  red  blood. 


54  OUTLINES  OF  LOGIC. 

5.  Disjunctive  Syllogisms. 

1.  DEFINITION. — A  disjunctive  syllogism  is  a  syllogism 
which  has  for  its  major  premise  a  disjunctive  judgment. 

The  minor  premise  and  the  conclusion  are  categorical 
judgments. 

2.  RULES. —  The  general  rule  governing  all  disjunctive 
syllogisms  is : 

If  one  or  -more  alternatives  be  affirmed,  the  rest  must  be 
denied,  and  if  one  or  more  alternatives  be  denied,  the  rest 
must  be  affirmed. 

This  rule  follows  immediately  from  the  law  of  excluded 
middle. 

3.  MOODS. —  There  are  two  moods,  viz.,  modus  ponendo 
tollens  (the  mood  which  by  affirming  denies)  and  modus 
tollendo  ponens  (the  mood  which  by  denying  affirms),  ac- 
cording as  the  minor  premise  is  affirmative  or  negative. 

a)  Modus  ponendo  tollens. —  The  general  form  of  this 
mood  is  A  is  either  B  or  not-B. 

A  is  B. 

.*.  A  is  not  not-B. 
For  instance, 

A  triangle  is  either  right-angled,  acute-angled,  or 

obtuse-angled. 

This  triangle  is  right-angled. 

.*.  This  triangle  is  neither  acute -angled  nor  obtuse- 
angled. 


UNIVERSITY 

OF 


SYLLOGISMS.  55 


fy  Modus  tollendo  ponens.  —  The  general  form  of  this 

mood  is 

A  is  either  B  or  not-B. 

A  is  not  not-B. 
.«.  A  is  B. 

For  instance, 

A  triangle  is  either  right-angled,  acute-angled,  or 

obtuse-angled. 
This    triangle    is    neither  right-angled    nor    acute- 

angled. 
.«.  This  triangle  is  obtuse  angled. 

6.  Dilemma. 

A  dilemma  is  a  syllogism  having  for  its  major  premise 
a  hypothetical  judgment  and  for  its  minor  premise  a  dis- 
junctive judgment. 

There  are  several  different  forms  of  the  dilemma.  We 
will  only  give  one  of  the  more  common  forms,  in  which 
the  major  premise  is  a  hypothetical  judgment  whose  con- 
sequent is  disjunctive.  This  form  of  the  dilemma  may  be 

stated  thus  : 

If  A  is,  either  B  or  C  is. 

Now  neither  B  nor  C  is. 
.*.  A  is  not. 

This  is  in  fact  a  destructive  hypothetical  syllogism.     All 


56 


OUTLINES  OF  LOGIC. 


possible  alternatives  of  the  consequent  are  denied,  there- 
fore the  antecedent  must  also  be  denied.     For  instance, 

If  this  triangle  is  not  right-angled,  it  must  be  either 

obtuse-angled  or  acute-angled. 
Now  it  is  neither  obtuse-angled  nor  acute-angled. 
.*.  It  must  be  right-angled. 

7.  Compound  Syllogisms. 

A  series  of  syllogisms  combined  together  in  such  a  man- 
ner that  the  conclusion  of  the  first  is  taken  as  a  premise 
of  the  second  and  so  on  is  called  a  compound  syllogism  or 
a  poly -syllogism. 

When  the  conclusion  of  one  syllogism  is  used  as  a 
premise  of  another  syllogism,  the  former  syllogism  is 
called  &  pro-syllogism  and  the  latter  an  epi-syllogism.  The 
conclusion  of  a  pro-syllogism  may  be  either  the  major  or 
the  minor  premise  of  the  epi-syllogism,  as  is  seen  by  the 
following  examples  : 

1.  2. 

All  C  are  D.          /^  N.  All  B  are  C. 

All  B  are  C.       /    /^~ "\\\        A11  A  are  R 
.-.  All  B  are  D.     /  /  //^~^\\\  \  .'.  All  A  are  C. 


All  B  are  D. 
All  A  are  B. 
All  A  are  D. 


All  C  are  D. 

All  A  are  C. 

.-.  All  A  are  D. 


SYLLOGISMS.  57 


For  A,  B,  C,  and  D  let  us  take  the  terms  square,  paral- 
lelogram, quadrilateral,  and  figure,  and  we  have  the  fol- 
lowing compound  syllogisms : 

1. 

All  quadrilaterals  are  figures. 
All  parallelograms  are  quadrilaterals. 
/.  All  parallelograms  are  figures. 

All  parallelograms  are  figures. 
All  squares  are  parallelograms. 
,\  All  squares  are  figures. 

2. 

All  parallelograms  are  quadrilaterals. 
All  squares  are  parallelograms. 
.*.  All  squares  are  quadrilaterals. 

All  quadrilaterals  are  figures. 
All  squares  are  quadrilaterals. 
.•.  All  squares  are  figures. 

8.  Abridged  Syllogisms. 

An  abridged  syllogism  is  a  syllogism  (either  simple  or 
compound)  in  which  one  or  more  of  the  premises  is  sup- 


This  is  the  usual  form  of  an  argument.  Perfectly 
formal  syllogisms  are  very  seldom  met  with.  But  in 
order  that  an  argument  which  has  not  the  form  of  a  per- 
fect syllogism  may  be  valid  it  must  be  capable  of  being 


58  OUTLINES  OF  LOGIC. 

put  into  the  form  of  regular  syllogisms.  It  should  also 
be  observed  that,  though  one  or  more  premises  may  be 
suppressed,  no  term  must  be  wanting. 

The   different  kinds  of  abridged  syllogisms  which  we 
will  consider  are : 

1.  The  Enthymeme. 

2.  The  Epichirema. 

3.  The  Sorites. 

1.  ENTHYMEME. — An  enthymeme  is  an  abridged  simple 
syllogism  in  which  one  or  both  of  the  premises  is  sup- 


The  ethymeme  is  of  two  kinds. 

a)  Either  the  major  or  the  minor  premise  is  suppressed. 
For  instance, 

The  square  is  a  parallelogram. 
.•.  The  opposite  angles  of  a  square  are  equal. 

All  men  are  mortal. 
/.  Napoleon  is  mortal. 

In  the  first  example  the  major  premise,  and  in  the  sec- 
ond the  minor  premise  is  suppressed. 

b)  Both  premises  are  suppressed,  the  middle  term  being 
included  in  the  conclusion.     For  instance, 

The  square,  being  a  parallelogram,  has  the  opposite 

sides  equal. 

The  enthymeme  has  very  often  the  form  of  a  sentence 
consisting  of  two  propositions,  united  by  the  conjunction 


SYLLOGISMS.  59 


'because.  Thus,  Napoleon  is  mortal  because  he  is  a  man  is 
really  an  enthymeine.  It  can  easily  be  put  into  the  form 
given  above. 

2.  EPICHIREMA. — An  epichirema   is  an  abridged  com- 
pound syllogism  is  which  one  or  more  of  the  premises  are 
enthymemes. 

For  instance, 

All  minerals,  being  material  bodies,  have  weight. 
Gold,  being  a  metal,  is  a  mineral. 
.•.  Gold  has  weight. 

This  may  be  put  into  the  regular  syllogistic  form  as  fol- 
lows : 

All  material  bodies  have  weight. 

1.  All  minerals  are  material  bodies. 
.*.  All  minerals  have  weight. 

All  metals  are  minerals. 

2.  Gold  is  a  metal. 
.•.  Gold  is  a  mineral. 

All  minerals  have  weight. 

3.  Gold  is  a  mineral. 
.•.  Gold  has  weight. 

3.  SORITES. — A  sorites  or  chain-argument  is  an  abridged 
poly -syllogism  consisting  of  three  or  more  simple  premises. 

There  are  two  kinds  of  sorites,  the  Aristotelian  and  the 
Goclenian,  the  former  having  been  invented  by  Aristotle 


60 


OUTLINES  OF  LOGIC. 


and  the  latter  by  Godenius.     These  two  kinds  of  sorites 
may  be  stated  in  the  following  way : 


Aristotelian. 

All  A  are  B. 
All  B  are  C. 
All  C  are  D. 
All  D  are  E. 
•.  All  A  are  E. 


Godenian. 

All  D  are  E. 
All  CUre  D. 
All  B  are  C. 
All  A  are  B. 
.-.  All  A  are  E. 


In  the  Aristotelian  sorites  the  predicate  of  one  premise 
becomes  the  subject  of  the  next,  and  the  conclusion  has 
for  its  subject  the  subject  of  the  first  premise,  and  for  its 
predicate  the  predicate  of  the  last  premise. 

In  the  Godenian  sorites  the  order  is  reversed.  The 
subject  of  one  premise  becomes  the  predicate  of  the  next, 
and  the  conclusion  has  for  its  subject  the  subject  of  the 
last  premise  and  for  its  predicate  the  predicate  of  the  first 
premise. 

In  the  Aristotelian  sorites  we  go  from  the  term  of  least 
extent  to  the  term  of  greatest  extent,  and  in  the  Godenian 
sorites  from  the  term  of  greatest  extent  to  the  term  of 
least  extent.  Therefore  the  former  is  also  called  an  as- 
cending sorites  and  the  latter  a  descending  sorites. 


SYLLOGISMS.  61 


EXAMPLES. 

Aristotelian  sorites. 
All  flies  are  insects. 
All  insects  are  invertebrates. 
All  invertebrates  are  animals. 
All  animals  are  organic  beings. 
.*.  All  flies  are  organic  beings. 

Goclenian  sorites. 
All  animals  are  organic  beings. 
All  invertebrates  are  animals. 
All  insects  are  invertebrates. 
All  flies  are  insects. 
.».  All  flies  are  organic  beings. 

In  regard  to  the  quality  and  quantity  of  the  premises, 
it  should  be  observed  that  in  the  Aristotelian  sorites  the 
only  premise  that  may  be  particular  is  the  first,  and  the 
only  one  that  may  be  negative  is  the  last.  The  Aristo- 
telian sorites  given  above  may  be  put  into  the  syllogistic 
form  as  follows : 

1.  2.  3. 

B  is  C  C  is  D  D  is  E 

A  is  B  A  is  C  A  is  D 

.\  A  is  C  .-.  A  is  D  .«.  A  is  E 

The  simple  syllogisms  of  which  the  sorites  is  composed 
are  all  in  the  first  figure,  and  in  this  figure  the  major 
premise  must  be  universal  and  the  minor  premise  affirma- 


62  OUTLIVES  OF  LOGTC. 

tive.  Hence  the  first  premise  of  the  sorites,  being  the 
only  minor  premise  expressed,  is  the  only  one  that  may 
be  particular. 

Again,  the  last  premise  of  the  sorites  is  the  only  one 
that  may  be  negative.  For  if  any  other  be  negative,  the 
conclusion  of  the  corresponding  simple  syllogism  would 
be  negative,  and  as  this  conclusion  is  to  be  used-  as  the 
minor  premise  of  the  next  syllogism,  we  would  have  a 
syllogism  in  the  first  figure  having  a  negative  minor 
premise,  which  is  contrary  to  the  rule. 


CHAPTER  Y. 


FALLACIES. 

1.   Definition  of  Fallacy. 

A  fallacy  is  an  argument  which  at  first  sight  appears 
to  be  valid,  but  in  reality  violates  the  rules  of  the  syllogism. 

2.   Classification  of  Fallacies. 

Fallacies  are  usually  divided  into  two  classes :   logical 
fallacies  and  material  fallacies. 

a)  A  logical  fallacy  is  one  in  which  the  premises  are 
insufficient  or  where  the  conclusion  does  not  follow  from 
the  premises. 

b)  A  material  fallacy  is  one  in  which  the  premises  are 
sufficient  for  the  conclusion,  bat  in  which  either  the  truth 
of  the  premises  remains  to  be  proved  or  the  conclusion  is 
irrelevant  to  the  point  that  is  to  be  demonstrated. 

A  material  fallacy  is  not  a  fallacy  in  the  form,  but  in 
the  subject-matter.  To  decide  whether  the  premises  are 
true  or  not,  is  something  that  logic  cannot  do.  The  sub- 
ject of  material  fallacies  is  therefore  one  with  which  logic 
is  only  indirectly  concerned. 

(63) 


64  OU 7 LINES  OF  LOGIC. 

3.  Logical  Fallacies. 

Logical  fallacies  may  be  divided  into  purely  logical  and 
semi-logical. 

Of  purely  logical  fallacies,  including  all  the  distinct  vio- 
lations of  the  syllogism,  the  following  may  be  mentioned  : 

1.  Fallacy  of  Four  Terms. 

2.  Fallacy  of  Undistributed  Middle. 

3.  Fallacy  of  Illicit  Process  of  either  Major  or  Minor  Term. 

4.  Fallacy  of  Negative  Premises. 

5.  Fallacy  of  Particular  Premises. 

All  these  fallacies  are  explained  in  the  chapter  treating 
of  syllogisms.  We  will  only  give  the  following  examples : 

A  is  B. 

1.      C  is  D. 

,-.  D  is  A. 

Here  we  have  no  middle  term  or  medium  of  comparison 
between  A  and  D.  Hence  in  order  to  compare  A  and  D 
two  syllogisms  are  required,  one  for  comparing  A  and  C 
with  B  and  the  other  for  comparing  A  and  C  with  D. 

All  birds  are  vertebrates. 

2.  All  fishes  are  vertebrates. 
.%  All  fishes  are  birds. 

All  insects  are  animals. 

3.  No  dogs  are  insects. 
.*.  No  dogs  are  animals. 


FALLACIES.  65 


No  birds  are  quadrupeds. 

4.  No  horses  are  birds. 

.*.  No  horses  are  quadrupeds. 

Some  flowers  are  blue. 

5.  Some  flowers  are  red. 

/.  Some  red  tilings  are  blue. 

Of  semi-logical  fallacies  the  more  common  are : 

1.  Fallacy  of  Equivocation, 

2.  Fallacy  of  Composition. 

3.  Fallacy  of  Division. 

4.  Fallacy  of  Accident. 

5.  Converse  Fallacy  of  Accident. 

6.  Fallacy  of  Many  Questions. 

7.  Fallacy  of  Amphibology. 

8.  Fallacy  of  Positive  and  Negative  Intention. 

1.  FALLACY  OF  EQUIVOCATION. —  This  fallacy  consists  in 
using  a  term  in  two  different  senses. 

In  most  cases  it  is  the  middle  term  that  is  used  in  two 
different  significations  in  the  premises.  In  such  a  case  the 
fallacy  is  usually  called  a  fallacy  of  ambiguous  middle. 
The  fallacy  of  equivocation  is,  in  reality,  a  fallacy  of  four 
terms,  as  is  easily  seen  by  substituting  some  other  expres- 


60  OUTLINES  OF  LOGIC. 

sion  for  the  ambiguous  term  in  each  premise.     For  in- 
stance, 

No  designing  person  ought  to  be  trusted. 
Engravers  are  designers. 
.*.  Engravers  ought  not  to  be  trusted. 

A  ball  is  a  round  body. 
He  attended  the  ball. 
.*.  He  attended  a  round  body. 

2.  FALLACY  OF  COMPOSITION. — This  fallacy  consists  in 
using  the  middle  term  distributively  in  the  major  premise 
and  collectively  in  the  minor  premise. 

For  instance, 

Five  and  three  are  two  numbers. 
Eight  is  five  and  three. 
.*.  Eight  is  two  numbers. 

3.  FALLACY  OF  DIVISION. — This  fallacy  consists  in  using 
the  middle  term  collectively  in  the  major  premise  and  dis- 
tributively in  the  minor  premise. 

For  instance, 

Eight  is  one  number. 
Five  and  three  are  eight. 
/.  Five  and  three  are  one  number. 

All  the  apples  in  the  garden  are  worth  one  hun- 
dred dollars. 

This  is  one  of  the  apples  in  the  garden.  % 

.*.  This  apple  is  worth  one  hundred  dollars. 


FALLACIES.  67 


4.  FALLACY  OF  ACCIDENT. — This  fallacy  consists  in  as- 
serting of  something  described  by  some  accidental  pecul- 
iarity what  is  true  only  of  its  substance. 

For  instance, 

What  yon  bought  yesterday  you  eat  to-day. 
You  bought  raw  meat  yesterday. 
.*.  You  eat  raw  meat  to-day. 

We  do  not  buy  meat  because  it  is  raw,  but  because  it  is 
meat.  That  the  meat  is  raw  is  only  an  accidental  prop- 
erty. 

5.  CONVERSE  FALLACY  OF  ACCIDENT. —  This  fallacy  con- 
sists in  arguing  from  a  special  case  to  a  general  one. 

For  instance, 

Alcohol  acts  as  a  poison  when  used  in  excess. 
.*.  Alcohol  is  always  a  poison. 

6.  FALLACY  OF  MANY  QUESTIONS. — This  fallacy  consists 
in  combining  two  or  more  questions  into  one  to  which  a 
single  answer  cannot  be  given. 

Thus,  if  a  man  who  has  never  used  tobacco  is  asked  If 
he  has  given  up  smoking,  he  can  neither  answer  the  ques- 
tion affirmatively  nor  negatively.  This  question  would 
namely  imply  that  he  did  smoke.  This  fallacy  arises  from 
the  fact  that  though  only  one  question  is  expressed,  two  or 
more  questions  are  implied. 


68  OUTLINES  OF  LOGIC. 

7.  FALLACY  OF  AMPHIBOLOGY. — This  fallacy  consists  in 
ambiguity  in  the  grammatical  structure  of  a  sentence,  by 
which  it  may  have  two  or  more  different  meanings. 

Thus  a  word  may  be  used  so  as  to  leave  it  ambiguous 
whether  it  is  subject  or  predicate,  or  the  reference  of  a 
pronoun  or  an  adverb  may  be  ambiguous.  For  instance, 

He  likes  me  better  than  you. 

We  also  get  salt  from  the  ocean,  which  is  very 
useful  to  man. 

He  promised  his  father  to  help  his  friends. 

8.  FALLACY   OF    POSITIVE  AND  NEGATIVE    INTENTION.— 
This  fallacy  consists  in  using  certain  negative  words,  as 
no  and  nothing,  in  two  different  senses. 

For  instance, 

No  cat  has  two  tails. 
Every  cat  has  one  tail  more  than  no  cat. 
.*.  Every  cat  has  three  tails. 

Nothing  is  better  than  happiness. 
Bread  is  better  than  nothing. 
/.  Bread  is  better  than  happiness. 

4.   Material  Fallacies. 

Of  material  fallacies  the  more  common  are : 
1.  Begging  the  Question  (Petitio  principii). 


FALLACIES.  69 


2.  Fallacy  of  False  Cause  (Non  causa  pro  causa). 

3.  Fallacy  of  Irrelevant  Conclusion  (Ignoratio  elenchi). 

1.  BEGGING    THE    QUESTION. — This  fallacy   consists   in 
using  as  a  premise  either  the  conclusion  itself  or  some  con- 
sequence of  the  conclusion  which  is  to  be  established. 

Another  name  for  this  kind  of  fallacy  is  arguing  in  a 
circle  (circulus  in  demonstrando).  Thus  we  argue  in  a 
circle  if  we  try  to  prove  the  existence  of  God  in  the  fol- 
lowing way : 

The  Scriptures  must  be  true,  as  they  are  the  word 
of  God. 

The  Scriptures  declare  that  God  exists. 
.•.  God  exists. 

Here  we  prove  that  God  exists  from  the  truth  of  the 
Scriptures  and  prove  the  truth  of  the  Scriptures  from  the 
fact  that  they  are  the  word  of  God,  which  evidently  im- 
plies that  we  take  for  granted  what  is  to  be  proved,  namely, 
that  God  exists. 

2.  FALLACY  OF  FALSE  CAUSE.  —  This  fallacy  consists  in 
assigning  as  a  cause  something  that,  in  reality,  has  noth- 
ing to  do  with  the  conclusion. 

If  one  event  occurs  shortly  before  another  event  or 
they  occur  at  the  same  time,  and  if  we  take  the  mere  con- 
junction of  the  two  events  as  a  satisfactory  proof  that  one 
is  the  cause  of  the  other,  we  commit  a  fallacy  of  false 


70  OUTLINES  OF  LOGIC. 

cause.     Two  events  may  be  simultaneous  without  having 
the  least  relation. 

3.  FALLACY  OF  IRRELEVANT  CONCLUSION. — This  fallacy 
consists  in  arriving  at  a  conclusion  different  from  the  one 
that  is  to  be  established. 

Suppose  we  had  to  prove  that  all  the  angles  of  a  tri- 
angle are  together  equal  to  two  right  angles,  and  we' only 
proved  that  they  cannot  be  less  than  two  right  angles. 
That  would  be  a  fallacy  of  irrelevant  conclusion,  for  the 
proposition  would  not  be  proved  before  we  had  also  proved 
that  the  angles  cannot  be  more  than  two  right  angles. 

The  fallacy  of  irrelevant  conclusion  is  one  of  the  most 
common  of  the  material  fallacies,  and  is  known  under 
various  names.  Of  the  more  common  forms  of  this  kind 
of  fallacy  the  following  two  may  be  mentioned : 

a)  Argumentum  ad  hominem,  which  consists  in  making 
an  appeal   to  the  vanity  or  prejudice  of  our  opponent  so 
as  to  make  him  blind  to  the  unreasonableness  of  the  argu- 
ment. 

b)  Argumentwn  ad  populum,  which   differs   from   the 
former  fallacy  only  in  being  addressed  to  a  body  of  people 
instead  of  one  individual. 

5.    Paralogisms  and  Sophisms. 

Fallacies  may  also  be  divided  into  paralogisms  and 
sophisms. 


FALLACIES.  71 


1.  A  paralogism  is  an  undesigned  fallacy,  the  person 
that  commits  it   being   unconscious  of  the  falsity  of  his 
argument. 

2.  A  sophism  is  a  fallacy  which  is  consciously  used  to 
deceive. 


CHAPTER  VI. 


METHOD. 

1.    Science. 

1.  DEFINITION  OF  SCIENCE. — Science  is  classified  knowl- 
edge. 

A  person  may  have  learned  a  good  many  facts  about  a 
certain  group  of  objects  or  phenomena,  but  in  order  that 
his  knowledge  may  be  entitled  to  the  name  of  scientific 
knowledge,  the  facts  must  be  arranged  according  to  cer- 
tain principles  and  the  relation  between  them  clearly  un- 
derstood. 

Scientific  knowledge  does  not  differ  in  kind  from  com- 
mon knowledge,  as  the  powers  used  in  acquiring  knowl- 
edge, whether  it  be  common  or  scientific,  must  obviously 
be  the  same.  They  differ  only  in  degree  of  accuracy. 

2.  REQUISITES  OF  A  SCIENCE. — The  requisites  of  a  science 
are: 

a)  All  statements  made  must  be  true. 

b)  A  science  should  be  as  general  as  possible;   i.  e.,  the 
process  of  generalization  should  be  carried  as  far  as  possi- 
ble. 

(72) 


METHOD.  73 


c)  In  every  science  there  should  be  a  certain  order,  and  a 
necessary  connection  between  the  various  elements  of  the 
science. 

d)  The  number  of  facts  ascertained  should  he  as  great 
as  possible. 

3.  AXIOMS.— An  axiom  is  a  self-evident  and  intuitively 
true  proposition. 

The  truth  of  an  axiom  cannot  and  need  not  be  demon- 
strated by  any  simpler  propositions. 

The  ultimate  principles  of  all  deductive  sciences  are 
axioms,  which  form  the  basis  on  which  all  the  demonstra- 
tions of  those  sciences  are  founded.  As  examples  of  axi- 
oms we  may  mention  the  following  two : 

The  whole  is  greater  than  its  parts. 

Things  that  are  equal  to  the  same  thing  are  equal 
to  each  other. 

2.   Deduction  and  Induction. 

1.  DEFINITION  OF  METHOD. — Method  is  a  certain  mode 
of  procedure  for  arriving  at  a  certain  result. 

Method  must  be  used  in  all  sciences,  though  the  kind  of 
method  which  is  to  be  used  will  be  different  for  different 
sciences. 

The  methods  used  in  science  may  be  classified  under 
the  two  heads,  deduction  and  induction. 


74:  OUTLINES  OF  LOGIC. 

2.  DEDUCTION. — Deduction  is  the  process  of  deriving  a 
particular  truth  from  a  general  truth. 

In  the  deductive  method  we  proceed  from  the  general 
to  the  particulars  which  are  embraced  in  it. 
For  instance, 

All  insects  are  animals. 
All  butterflies  are  insects. 
.-.  All  butterflies  are  animals. 

Here  we  first  state  a  general  truth,  something  that  is 
true  about  all  insects,  namely,  that  they  are  animals.  Then 
we  proceed  to  analyze  this  general  truth  into  the  partic- 
ulars it  embraces,  and  finally  we  reach  a  conclusion  con- 
cerning one  of  the  particulars,  namely,  butterflies. 

The  deductive  method  is  also  called  the  analytic  method. 

3.  INDUCTION. — Induction  is  the  process  of  deriving  gen- 
eral truths  from  particular  truths. 

In  the  inductive  method  we  proceed  from  the  observa- 
tion of  particular  truths  or  facts  to  the  establishment  of 
general  laws.  As  an  example  of  inductive  reasoning  we 
give  the  following : 

By  observations  we  know  that  Mercury,  Venus,  the 
Earth,  Mars,  Jupiter,  Saturn,  Uranus  and  Nep- 
tune move  around  the  sun  in  elliptic  orbits. 

Hence  all  the  planets  move  around  the  sun  in 
elliptic  orbits. 


METHOD.  75 


The  inductive  method  is  also  called  the  synthetical 
method. 

Induction  is  of  two  kinds:  Perfect  induction  and  im- 
perfect induction. 

a)  The  induction  is  perfect  when  all  the  particular  cases 
have  been  examined. 

For  instance, 

Mercury,  Yenus,  the  Earth,  Mars,  etc.,  move  in  el- 
liptic orbits  around  the  sun. 

Hence  all  the  known  planets  move  around  the  sun 
in  elliptic  orbits. 

In  the  conclusion  we  affirm  something  only  of  the  par- 
ticular cases  that  have  been  examined.  We  do  not  say 
that  all  planets  move  in  elliptic  orbits  around  the  sun,  but 
only  all  the  known  planets.  The  conclusion  must  there- 
fore be  certain. 

Perfect  induction  always  leads  to  a  necessary  and  cer- 
tain conclusion. 

&)  The  induction  is  imperfect  when  we  have  examined 
only  some  of  the  particular  cases  and  from  them  infer  a 
general  law. 

In  the  first  example  given  above  we  assert  of  all  planets 
something  that  has  been  found  to  be  true  of  all  the  known 
planets.  Hence  we  infer  that  if  some  new  planet  would 
be  discovered  it  would  most  likely  move  in  an  elliptic 


76  OUTLINES  OF  LOGIC. 

orbit  around  the  sun  like  the  planets  that  are  now  known. 
This  conclusion  is  very  probable,  but  not  certain. 

Imperfect  induction  can  never  lead  to  a  certain  and 
necessary  conclusion,  but  only  to  a  probable  conclusion. 

3.   Definition. 

1.  DEFINITION  DEFINED. — To  define  a  thing  is  to  give 
those  attributes  by  which  it  differs  from  all  other  things, 
and  the  process  is  called  logical  definition. 

To  define  something  means  to  state  what  it  is,  or  to 
distinguish  it  from  all  other  things.  It  is  not  necessary, 
however,  to  enumerate  all  the  attributes  belonging  to  the 
thing  which  is  to  be  defined,  but  only  the  essential  attri- 
butes. The  essential  attributes  are  the  genus  and  the  dif- 
ferentia. 

a)  By  the  genus  is  meant  the  next  higher  genus  of  which 
the  thing  to  be  defined  is  a  species. 

ft)  By  the  differentia  is  meant  those  specific  characters 
~by  which  the  thing  to  he  defined  differs  from  all  other  spe- 
cies of  the  same  genus. 

Definition  thus  consists  in  giving  the  genus  and  the 
differentia  of  the  thing  to  be  defined.  A  definition  has 
the  form  of  a  categorical  judgment,  of  which  the  subject 
is  the  thing  to  be  defined,  and  the  predicate  the  genus  and 
the  differentia. 

Suppose  we  want  to  define  an  equilateral  triangle.  An 
equilateral  triangle  is  a  species  of  the  genus  triangle,  and 


METHOD.  77 


differs  from  all  other  triangles  in  having  the  three  sides 
equal.  Hence  the  genus  is  triangle  and  the  differentia 
having  the  three  sides  equal.  The  definition  of  an  equilat- 
eral triangle  will  then  be : 

An  equilateral  triangle  is  a  triangle 

genus. 
having  its  sides  equal. 


It  should  be  observed  that  it  is  essential  for  a  logical 
definition  that  the  genus  should  be  the  next  higher  genus. 
Hence  the  following  definition  is  not  correct: 

An  equilateral  triangle  is  a  plane  figure  having  its 

sides  equal. 

Plane  figure  is  not  the  next  higher  genus. 
We  will  give  two  more  examples  of  definitions,  viz. : 
Man  is  a  rational  animal. 

differentia     genus 
A  parallelogram  is  a  quadrilateral 

genus 

whose  opposite  sides  are  parallel. 

differentia. 

2.  RULES  FOR  DEFINITION. — In  definition  the  following 
rules  should  be  observed  : 

I.  The  definition  should  be  adequate,  i.  e.,  neither  too 
wide  nor  too  narrow. 

a)  The  definition  is  too  wide  if  the  predicate  has  greater 


78  OUTLINES  OF  LOGIC. 

extent  than  the  subject,  i.  e.,  if  it  includes  other  things  be- 
sides those  that  are  to  be  defined. 

For  instance, 

« 

A  bird  is  an  animal  that  has  a  backbone. 

This  definition  is  too  wide  because  also  fishes,  reptiles 
and  mammals  have  a  backbone. 

Man  is  a  rational  being. 

This  is  also  too  wide,  because  rational  being  also  ill- 
dudes'  God. 

b)  The  definition  is  too  narrow  if  the  subject  lias  a 
greater  extent  than  the  predicate,  i.  e.,  if  it  excludes  some 
of  the  things  that  are  to  be  defined. 

For  instance, 

A  triangle  is  a  figure  having  three  equal  sides. 

This  definition  is  too  narrow,  because  all  isosceles  and 
scalene  triangles  are  excluded. 

A  bird  is  a  feathered  animal  that  sings. 

This  is  also  too  narrow.      Some  birds  do  not  sing. 

The  test  of  an  adequate  definition  is  that  it  may  be  both 
simply  converted  and  converted  by  contraposition.  If  the 
definition  is  too  wide,  it  cannot  be  simply  converted.  If 
it  is  too  narrow,  it  cannot  be  converted  by  contraposition. 

II.  The  definition  should  not  contain  the  term  which  is 
to  be  defined. 

The  violation  of  this  rule  is  called  defining  in  a  circle. 
We  thus  define  in  a  circle  if  we  define  law  as  a  lawful 


METHOD.  79 


command,  because  we  use  in  the  definition   the  word  we 
want  to  define.     As  another  example  let  us  take 

Life  is  the  sum  of  the  vital  functions. 
Here  we  use  the  term  vital,  which  is  really  a  synonym 
of  the  term  to  be  defined,  and  which  only  can  be  explained 
by  the  term  life. 

III.  The  definition  should  he  affirmative. 

The  definition  should  state  what  a  thing  is,  and  not 
what  it  is  not.  Hence  the  following  definitions  are  unsat- 
isfactory : 

A  straight  line  is  a  line  no  portion  of  which  is 
curved. 

A  regular  polygon  is  one  that  is  not  irregular. 
Light  is  the  absence  of  darkness. 

IV.  In  definition  we  should  not  give  any  superfluous  or 
accidental  attributes. 

For  instance, 

A  pentagon  is  a  polygon  having  five  sides  and  five 
angles. 

This  definition  is  incorrect,  as  the  latter  attribute  is 
superfluous. 

A  parallelogram  is  a  quadrilateral  having  the  op- 
posite sides  parallel  and  having  the  opposite 
sides  and  angles  equal. 

Here  two  attributes  are  given  that  follow  from  the  par- 
allelism of  the  sides,  and  which  therefore  are  superfluous. 


80  OUTLINES  OF  LOGIC. 

A  horse  is  a  four-legged  animal  with  a  tail  and  a 

mane. 
Here  accidental  attributes  are  used. 

3.  NOMINAL  AND    REAL    DEFINITIONS. —  Definitions    are 
divided  into  nominal  and  real. 

a)  A  nominal  definition  is  one  which  explains  the  mean- 
ing of  the  term  which  is  used  as  the  name  of  the  thing. 

For  instance, 

A  phonograph  is  an  instrument  for  registering  and 

reproducing  sound. 
A  telephone  is  an  instrument  for  conveying  sound 

to  a  great  distance. 

b)  A  real  definition  is  one  which  defines  the  thing  itself. 
Thus  a  real  definition  of  phonograph  would  be  a  treatise 

on  the  construction  and  use  of  that  instrument. 

In  all  scientific  investigations  it  is  the  aim  to  obtain 
real  definitions,  but  for  many  practical  purposes  nominal 
definitions  will  be  sufficient. 

4.  DESCRIPTION. — By  description  is  meant  an  enumera- 
tion of  all  the  properties  of  a  thing. 

A  description  of  an  elephant,  for  instance,  would  thus 
consist  in  the  enumeration  of  all  the  properties  belonging 
to  elephants.  In  definition  we  give  only  the  essential  at- 
tributes of  a  thing.  In  description,  again,  we  may  use 
not  only  essential,  but  also  accidental  attributes.  The 


METHOD.  81 


natural-history  sciences  furnish  good  examples  of  descrip- 
tions. 

4.   Division. 

1.  DIVISION  DEFINED. — By  logical  division  is  meant  the 
process  of  dividing  a  genus  into  its  species  according  to  a 
certain  principle  of  division. 

For  instance, 

(  right-angled 

Triangles  may  be  divided  into  •<  acute-angled 

I  obtuse-angled. 

Here  the  genus  triangle  is  separated  into  its  three  spe- 
cies, and  the  basis  or  principle  of  division,  commonly  called 
the  fundamentum  divisionis,  is  the  size  of  the  angles. 

triangles 


Polygons  may  be  divided  into  -< 


quadrilaterals 
pentagons 


hexagons 

etc. 
Here  the  principle  of  division  is  the  number  of  sides. 

2.  DICHOTOMY. — If  a  genus  is  divided  into  two  species 
each  of  which  is  the  contradictory  of  the  other,  the  divi- 
sion is  commonly  called  dichotomy. 

For  instance, 

Animals  may  be  divided  into  {  vertebrates 

(  not-vertebrates. 

(  triangles 
Polygons  may  be  divided  into  < 

(  not-triangles. 


82  OUTLINES  OF  LOGIC. 

Although,  from  a  logical  point  of  view,  dichotomy  is  a 
perfect  division,  it  is  for  most  practical  purposes  not  very 
convenient. 

3.  RULES  FOR  DIVISION. — In  logical  division  the  follow- 
ing rules  should  be  observed : 

I.  In  division  there  should  be  only  one  principle  of  di- 
vision. 

Hence  the  following  divisions  are  not  correct : 

English 
French 

,.   .,    ,    .  German 

Books   are   divided   into  4 

Quarto 
Octavo 

etc. 

The  first  division  is  according  to  language  and  the  sec- 
ond according  to  size. 

isosceles 

equilateral 
Triangles  are  divided  into  -< 

right-angled 

acute-angled. 

The  first  division  is  according  to  the  relative  length  of 
the  sides  and  the  second  according  to  the  size  of  the  an- 
gles. 

Such  a  division  is  generally  called  a  cross-division. 

II.  The  principle  of  division  should  be  an  actual  attri- 
bute of  the  genus  which  is  to  he  divided. 


METHOD.  83 


III.  In  division  the  members  should  exclude  each  other, 
and  they  should  all  be  co-ordinate  or  of  the  same  rank. 

For  instance, 

triangles 

quadrilaterals 
parallelograms 
polygons  having  more 
than  four  sides. 


Polygons  may  be  divided  into 


Parallelograms  are  included  in  quadrilaterals,  and  con- 
sequently the  members  do  not  exclude  each  other. 

IV.  The  division  should  be  complete,  i.  e.,  the  sum  of 
the  species  should  be  equal  to  the  genus. 

Hence  no  species  must  be  left  out.     For  instance, 

(  mammals 

Vertebrates  are  divided  into  •<  birds 

I  fishes. 
Here  reptiles  and  hatrachians  are  left  out. 

(  acute-angled 
Triangles   are  divided   into    \ 

(  right-angled. 

Here  obtuse-angled  triangles  are  left  out. 

V.  In  division  v}e  should  proceed  from  proximate  gen- 
era to  proximate  species. 

We  should  not  proceed  from  a  high  genus  to  a  low  spe- 
cies, but  from  the  genus  to  the  next  lower  species. 


84  OUTLINES  OF  LOGIC. 

In  the  following  division  this  rule  is  violated : 

horses 
dogs 


Vertebrates  are  divided  into  -< 


eagles 
lions 
etc. 


A  logical  division  of  vertebrates  would  be  into 

C  mammals 
birds 
fishes 

batrachians 
reptiles. 

Each  of  these  species  may  further  be  divided  and  sub- 
divided until  we  reach  the  lowest  species. 

4.  PARTITION. — By  partition  is  meant  the  separation  in 
thought  of  the  physical  parts  of  which  an  individual  ob- 
ject is  composed. 

For  instance, 

Water  is  composed  of  oxygen  and  hydrogen. 

A  plant  may  be  divided  into  root,  stem,  leaves,  etc. 

This  mode  of  separating  an  object  into  its  constituent 
parts  is  something  with  which  logic  is  not  directly  con- 
cerned, and  should  not  be  confounded  with  logical  divi- 
sion. 


METHOD.  85 


5.   Demonstration. 

1.  DEMONSTRATION   DEFINED. — Demonstration  is  an  act 
of  reasoning  by  which  the  truth  of  a  proposition  is  estab- 
lished as  a  consequence  of  other  truths. 

In  every  demonstration  we  notice: 

a)  The  proposition  that  is  to  be  proved. 

b)  The  premises  or  grounds  of  proof. 

c)  The  necessary  connection  between  the  different  parts 
of  the  demonstration. 

The  premises  are  either  definitions,  axioms,  or  previously 
established  propositions, 

2.  RULES  FOR  DEMONSTRATION. — For  demonstration  we 
have  the  following  rules : 

I.  No  proposition  must  be  used  as  a  premise  which  is 
not  known  to  be  true. 

II.  The  proposition  which  is  to  be  proved  must  not  be 
used  as  a  premise. 

III.  No  proposition  whose  truth  depends  on  the  truth 
of  the  proposition  which  is  to  be  proved  must  be  used  as  a 
premise. 

IY.   There  must  be  no  leaps  in  the  demonstration. 

Y.  We  must  not  prove  another  proposition  instead  of 
the  one  that  is  to  be  established. 

For  violations  of  the  rules  given  above  see  Chapter  Y 
(Fallacies). 


86  OUTLINES  OF  LOGIC. 

3.  CLASSIFICATION  OF  DEMONSTRATIONS.  —  Demonstra- 
tions are  divided  — 

I.  Into  direct  and  indirect. 
II.  Into  deductive  and  inductive. 
III.  Into  a  priori  and  a  posteriori. 

I.  a)  A  direct  demonstration  is  one  in  which  the  truth 
of  a  proposition  is  immediately  deduced  from  certain  other 
truths  that  have  already  been  established. 

b)  An  indirect  demonstration  is  one  in  which  the  truth 
of  a  proposition  is  established  by  proving  the  absurdity  of 
its  contradictory. 

In  a  direct  demonstration  we  give  the  reasons  why  the 
conclusion  must  be  true.  In  an  indirect  demonstration 
we  give  the  reasons  why  it  cannot  be  false.  In  an  in- 
direct demonstration  we  proceed  in  the  following  manner. 
We  make  a  supposition  contrary  to  the  conclusion  which 
is  to  be  proved.  From  this  supposition  we  deduce  a  series 
of  conclusions  until  we  arrive  at  a  conclusion  which  is  con- 
trary to  some  known  truth.  Then  by  modus  tollens  we 
conclude  from  the  falsity  of  the  consequent  to  the  falsity 
of  the  antecedent ;  that  is,  we  conclude  that  the  supposi- 
tion made  must  be  false,  as  it  leads  to  an  absurd  con- 
clusion. And  since  this  supposition  is  false,  its  contra- 
dictory, or  the  conclusion  which  is  to  be  established,  must 
be  true ;  because  of  two  contradictories  one  must  be  true 
and  the  other  false. 


METHOD. 


87 


We  give  the  following  example  of  an  indirect  demon- 
stration : 

Two  straight  lines  perpendicular  to  the  same  straight 
line  are  parallel. 


Let  the  two  straight  lines  AB  and  CD  be  both  per- 
pendicular to  AC ;  then  AB  is  parallel  to  CD. 

For  suppose  that  AB  is  not  parallel  to  CD.  Then  the 
two  lines  AB  and  CD  must  meet  at  some  point  if  they  be 
produced.  Let  them  meet  at  the  point  E.  Then  there 
will  be  two  perpendiculars,  EA  and  EC,  let  fall  from  the 
same  point  on  the  same  straight  line,  which  is  absurd. 
Therefore  the  two  lines,  AB  and  CD,  cannot  meet  if 
they  be  produced  ever  so  far.  Hence  the  two  lines  are 
parallel. 

II.  a)  A  deduction  demonstration  is  one  in  which  we 
proceed  from  the  whole  to  the  parts. 


88  OUTLINES  OF  LOGIC. 

b)  An  inductive  demonstration  is  one  in  which  we  pro- 
ceed from  the  parts  to  the  whole. 

In  a  deductive  demonstration  we  prove  that  something 
holds  true  of  the  whole,  and  then  conclude  that  it  must 
hold  true  of  every  part  or  individual  case  of  the  whole. 
In  an  inductive  demonstration,  again,  we  prove  that  some- 
thing holds  true  of  all  the  parts  or  individual  cases  and 
then  conclude  that  it  must  hold  true  of  the  whole. 

We  will  give  the  following  example  of  an  inductive 
demonstration : 

An  angle  inscribed  in  a  segment  is  'measured  by  half  the 
arc  included  between  its  sides. 

This  proposition  admits  of  three  cases: 
1st.  Let  the  centre  of  the  circle  be  on  one  of  the  sides 
of  the  angle. 

^  Draw  the  radius  OC.  Because 
Y  OC  is  equal  to  OB,  the  angle  OBC 
is  equal  to  the  angle  OCB ;  there- 
fore the  angles  OBC  and  OCB  are 
A-  together  double  the  angle  OBC. 
The  angle  AOC  is  equal  to  the  sum 
of  the  angles  OBC  and  OCB.  Hence  the  angle  AOC  is 
double  the  angle  OBC.  But  the  angle  AOC  is  measured 
by  the  arc  AC.  Hence  the  angle  ABC  is  measured  by 
half  the  arc  AC. 


METHOD, 


89 


2d.  Let  the  centre  of  the  circle  be  within  the  angle. 

Draw  the  diameter  BD.  By  the 
first  case  we  know  that  the  angle 
ABD  is  measured  by  half  the  arc  AD 
and  the  angle  DBC  by  half  the  arc 
DC.  Therefore  the  angle  ABC  is 
measured  by  half  the  sum  of  the  arcs 
AD  and  DC,  i.  e.,  half  the  arc  AC. 

3d.  Let  the  centre  be  w.thout  the  angle. 

/ 

Draw  the  diameter  BD.  By  the 
first  case  we  know  that  the  angle 
ABD  is  measured  by  half  the  arc 
AD,  and  the  angle  CBD  by  half  the 
arc  CD.  Therefore  the  angle  ABC 

*? ^-  is  measured  by  half  the  difference  of 

the  arcs  AD  and  CD,  i.  e.,  half  the  arc  AC. 

Hence  the  proposition  is  true  for  all  possible  cases,  and 
therefore  it  must  be  true  for  any  angle  inscribed  in  a  seg- 
ment. 

III.  a)  A  demonstration  a  priori  is  one  in  which  the 
premises  are  given  by  intuition. 

5)  A.  demonstration  a  posteriori  is  one  in  which  the 
premises  are  given  by  experience. 

In  mathematics,  for  instance,  all  the  relations  between 
quantities  are  established  by  a  chain  of  reasoning  which 
ultimately  depends  on  certain  a  priori  or  intuitive  princi- 


90  OUTLINES  OF  LOGIC. 

pies,  namely  the  ideas  of  space  and  number.  In  the  nat- 
ural sciences,  again,  the  arguments  are  mainly  a  posteriori, 
as  the  premises  are  given  by  experience. 

6.   Analogy. 

Reasoning  by  analogy  is  a  process  by  which  we  infer 
that  if  two  or  more  objects  are  similar  in  certain  respects, 
they  will  also  be  similar  in  other  respects. 

Reasoning  by  analogy  gives  only  a  probable  conclusion. 
The  degree  of  probability  depends  on  the  number  of  ob- 
served resemblances  and  the  importance  of  the  points  in 
which  the  objects  agree.  Hence  in  order  that  reasoning 
by  analogy  should  be  of  any  value,  the  attributes  that  are 
similar  should  be  as  many  as  possible  and  should  not  be 
accidental.  If  it  can  be  shown  that  one  or  more  of  the 
essential  attributes  of  the  first  object  is  incompatible  with 
some  essential  attribute  of  the  second  object,  the  argu- 
ment is  invalid. 

For  instance, 

By  observing  the  similarity  between  lightning  and 
electricity  in  many  respects,  Franklin  was,  by 
analogy,  led  to  the  conclusion  that  they  were 
identical. 

The  earth  and  the  planet  Mars  resemble  each  other 

in  many  respects. 
Hence  Mars  is  probably  inhabited. 


METHOD.  91 


7.   Hypothesis. 

A  hypothesis  is  a  supposition  made  to  accmmt  for  a  cer- 
tain group  of  phenomena. 

The  probability  of  a  hypothesis  depends  on  the  number 
of  facts  or  phenomena  that  may  be  explained  by  it.  The 
greater  the  number  of  phenomena  it  will  explain,  the  more 
we  are  justified  to  believe  the  hypothesis  to  be  right. 

As  examples  of  hypotheses  we  may  mention  Laplace's 
Nebular  Hypothesis  to  explain  the  formation  of  the  solar 
system,  and  the  Copernican  theory  of  the  solar  system. 

8.   Classification  of  Sciences. 

From  a  formal  point  of  view  the  sciences  are  usually 
divided  into  empirical  and  rational. 

The  difference  between  the  empirical  and  the  rational 
sciences  is  given  in  the  following  schedule : 
Data:  facts. 

Aim  :   the  establishment  of  general 
Empirical    -< 

laws. 

Method:   mainly  inductive. 
Data:   universal  principles. 

Aim:  the  establishment  of  particu- 
Rational    4 

lar  truths. 

Method:   mainly  deductive. 
Botany,  zoology,  chemistry  and  geology  are  examples 
of  empirical  sciences.     Mathematics  is  an  example  of  a 
rational  science. 


EXERCISES. 


CHAPTER  II. 

CLASSIFICATION   OF   CONCEPTS. 

1.  For  each  one  of  the  following  concepts  state  whether 
it  is  positive  or  negative,  absolute  or  relative,  concrete  or 
abstract : 

Book  t     Man  Daughter  . 

Father  House  Metal 

Weight  Darkness          Independence 

Holiness  Logic  Whiteness 

Unnatural  Light  Son 

Air  Resemblance    Animal 

Oblique-angled    Curved  Straight 

Being  Reason  Rational 

Figure  Triangle  God 

EXTENT  AND   CONTENT. 

1.  In  each  one  of  the  following  pairs  of  concepts  state 
which  concept  has  the  greater  extent,  and  which  has  the 
greater  content: 

Dog  JPlant  JMan 

Animal  ^(Tree  *(  Being 

(92) 


EXERCISES.  93 


{Heavenly  body  ( Element  ( Eagle 

Planet  6j  Metal  6(Bird 
(  Equitable  triangle            ( Fish  ( Rock 

(  Equiangular  triangle       (  Vertebrate      (  Granite 
|  Fly  (  Book  ( House 

( Insect  (  Dictionary       (  Brick  house 

2.  Arrange  the  following  terms  in  several  series  in  such 
a  manner  that  the  first  term  of  each  series  shall  have  the 
greatest  extent  and  the  last  term  the  least  extent. 


Salmon 

Plant 

Europeans 

Polygon 

Fish 

Square 

Man 

Animal 

Figure 

Apple  tree 

Vertebrate 

Rational  being 

Plane  figure 

Quadrilateral 

Phoenogam 

RELATION   OF   CONCEPTS. 

1.   State  the  relation   between  the  concepts  of  each  of 
the  following  groups : 

( Plant                              (  Polygon  ( Flies 

1  •]                                     2-1  3-1 

( Organic  being               (  Figure  ( Bees 

i  Square                            (  Man  j  European 

Parallelogram               j  American  ( Italian 

(Metal                              (Gold  (Salmon 

7  (Not-metal                     (Iron  9jFish 

( Bird                               ( Straight  ( Eagle 

|  Reptile                          ( Next-straight  ( Sparrow 


94  OUTLINES  OF  LOGIC. 

CHAPTER  III. 

CLASSIFICATION   OF  JUDGMENTS. 

State  the  logical  character  as  to  quality,  quantity,  rela- 
tion and  modality  of  each  of  the  following  judgments : 

All  triangles  are  figures. 
If  he  is  honest,  he  should  speak  the  truth. 
Triangles  are  divided  into  right-angled  and  oblique- 
angled. 

The  table  is  black. 
If  rain  has  fallen,  the  ground  is  wet. 
Napoleon  was  a  great  man. 
No  triangles  are  squares. 
Some  angles  are  obtuse. 
Some  horses  are  not  black. 
His  character  is  either  good  or  bad. 
Iron  is  an  element. 
Some  men  are  good. 
God  is  omniscient. 
Some  men  are  not  kings. 
This  horse  is  not  black. 
Some  triangles  are  equilateral. 
No  planets  are  self-luminous. 
Some  of  our  muscles  are  involuntary. 
New  York  is  a  city. 
Horses  are  vertebrate  animals. 


EXERCISES.  95 

IMMEDIATE   INFERENCE. 

1.  Which  of  the  four  judgments  J.,  /,  E  and  0  are  true 

or  false  when 

1.  A  is  true  5.  E  is  true 

2.  A  is  false  6.  E  is  false 

3.  /  is  true  7.  O  is  true 

4.  /  is  false  8.  0  is  false 

2.  Convert  the  following  judgments  : 

All  vertebrates  are  animals. 

Some  poisonous  things  are  plants. 

No  men  are  angels. 

Man  is  mortal. 

Some  persons  are  wise. 

Some  quadrupeds  are  not  horses. 

Some  birds  are  eagles. 

No  plants  are  animals. 

All  triangles  have  three  sides. 

No  triangles  are  quadrilaterals. 

3.  If  the  judgment 

Some  triangles  are  not  figures 

is  false,  how  could  you  prove  the  truth  of  the  judg- 
ment 

Some  triangles  are  figures? 

4.  How  can  you  conclude  from  the  falsity  of  the  judg- 

ment 

No  animals  living  in  water  are  fishes 
to  the  truth  of 

Some  fishes  live  in  water  \ 


OUTLINES  OF  LOGIC. 


5.  How  can  you  conclude  from  the  truth  of  the  judgment 

No  insects  are  vertebrates 
to  the  falsity  of 

Some  vertebrates  are  insects  ? 

6.  How  can  you  prove  the  falsity  of  the  judgment 

No  not-triangles  are  figures 
from  the  truth  of 

Some  figures  are  not  triangles  ? 


CHAPTER  IV. 
SIMPLE  SYLLOGISMS. 

Construct  syllogisms  from  the  terms  given  in  each  of 
the  following  moods : 

1st  Figure. 

(  P  =  animal  i  P  =  irrational 

Barbara  KM  —  bird  Celarent  •<  M  =  man 

(  S  —  eagle  (  S  —  American 

iP  =  mortal  C  P  =  square 

M  =  man  Ferio  \  M  =  triangle 

S  =  being  v  8  ==  equilateral 

figure 
2d  Figure. 

(  P  =  animal  (  P=insect 

Cesar  el  M=plant  Camestres\  M— animal 

(  S=grass  (  S=rock 


EXERCISES.  97 


(  P— trapezium  (  P=fixed  star 

Festino  ^  M=parallelogram  Baroko  •<  Myself  luminous 

\  S=quadrilateral  '  S=^heavenly 

body 

3d  Figure. 

P=man 
i"{  M=American        Disamis-{  M^vertebrate 


(  P=mortal  C  P  =  herb 

Datisi  -\  M=man  Felapton  •<  M  =  tree 

(  S— black  I  S  — plant 


(  P  —  having  feet  T  P  =  triangle 

Bokardo  vM  =  reptile  Ferison  *\  M  =  pentagon 

lS  =  animal  (  S  =  equilateral 


4th  Figure. 

(  P  =  granite  T  P  =  European 

Bramantip  •<  M  =  rock  Camenes  K  M  =  man 

\  S  =  inorganic  lS  = 


Dimaris  •<  M  =  butterfly 

I  S  =  insect  I  S  =  animal 

r  P  —  square 

Fresison  -s    M  =  hexagons 
L  S  =  equilateral 

—7 


0$  OUTLINES  OF  LOGIC. 

COMPOUND   SYLLOGISMS. 

1.  From  the  following  terms  construct  compound  syllo- 
gisms, epichiremata,  and  sorites : 

'  Organic  being 
Plant  Animal 

Phenogam  3  \   Vertebrate 

m ,  I    Reptile 

Oak  t  Cwodile 


Figure 

Plane  figure 

Man 
Polygon  4  -< 

European 
Triangle 

German 
^  Isosceles  triangle 

2.   Construct  enthymes  by  taking  any  three  consecutive 
terms  of  those  given  above. 


CHAPTER  V. 

FALLACIES. 

Point  out  the  fallacies  in  the  following  arguments 

1.      Some  plants  are  trees. 

Some  plants  are  grasses. 
.*.  Some  grasses  are  trees. 


EXERCISES.  99 


2.  Red  is  a  color. 
Blue  is  a  color. 

/.  Bine  is  red. 

3.  All  men  are  rational  beings. 
All  men  are  animals. 

/.  All  animals  are  rational  beings. 

4.  All  men  are  organic  beings. 
No  dogs  are  men. 

.'.  No  dogs  are  organic  beings. 

5.  All  moral  beings  are  accountable. 
No  brute  is  a  moral  being. 

.*.  No  brute  is  accountable. 

6.  Design  implies  a  designer. 

The  universe  abounds  in  design. 
.».  God  exists. 

7.  A  stone  is  a  body. 
An  animal  is  a  body. 
Man  is  an  animal. 

.».  Man  is  a  stone. 

8.  Nothing  is  better  than  wisdom. 
A  dime  is  better  than  nothing. 

.*.  A  dime  is  better  than  wisdom. 


100  OUTLINES  OF  LOGIC. 

9.      Metals  are  elements. 

Iron  is  a  metal. 
.*.  Iron  is  an  element. 

10.  If  this  medicine  is  of  any  value,  those  who  take 

it  will  improve  in  health. 
I  have  taken  it,  and  have  improved  in  health. 
/.  This  medicine  is  of  value. 

11.  Dickens's  Oliver  Twist  is  one  of  the  books  in 

the  book-store  of  my  friend. 
I  have  bought  Dickens's  Oliver  Twist. 
.•.  I  have  bought  one  of  the  books  in  my  friend's 
book-store. 

12.  His  books  are  worth  one  hundred  dollars. 

Shakespeare  is  one  of  his  books. 
,*.  Shakespeare  is  worth  one  hundred  dollars. 

13.  The  people  of  the  city  are  suffering  from  the 

yellow  fever. 

You  are  one  of  the  people  of  the  city. 
.•.  You  are  suffering  from  the  yellow  fever. 

14.  Light  is  contrary  to  darkness. 
Feathers  are  light. 

.*.  Feathers  are  contrary  to  darkness. 


EXERCISES.  101 


CHAPTER  VI. 


DEFINITION. 


1.   Define  the  following  terms,  and  point  out  the  genus 
and  the  differentia  in  each  definition : 


Element 

Capital 

Dictionary 

Nonagon 

Vertebrate 

Genus 

Logic 

Animal 

Species 

Science 

Man 

Hypothesis 

Syllogism 

Rhombus 

Circle 

Deduction 

Plant 

Straight  line 

Induction 

Parallelogran 

i    Judgment 

2.  What  rules  do  the  following  definitions  violate? 

1.  A   straight   line   is   one   no   portion   of   which   is 

curved. 

2.  A  rectangle  is  a  figure  having  four  right  angles. 

3.  A  trapezium  is  a  quadrilateral  having  the  oppo- 

site sides  parallel. 

4.  A  hexagon  is  a  figure  having  six  equal  sides. 

5.  A  mammal  is  an  animal  that  does  not  reproduce 

its  species  by  laying  eggs. 

6.  A  square  is  a  four-sided  figure  with  equal  sides. 

7.  Evil  is  that  which  is  not  good. 


102 


OUTLINES  OF  LOGIC. 


DIVISION. 

In  what  are  the  following  divisions  faulty  'i 


1.   Plants  are  divided  into 


Cryptogams 
Monopetalons 
Apetalous 
Polypetalous 


f  Men 
2.  Mankind  may  be  divided  into  J    Women 

I  Children 


3.  Birds  are  divided  into 


Sea-birds 

Sparrows 

Eagles 

Parrots 

Gallinaceous  birds 


4.  The    faculties    of    the    f  PercePtio« 

mind  are  divided  into  ]    ***#**&* 
I  Reason 

r  Grammars 
Dictionaries 

5.  Books  are  divided  into  •<    French 

German 
Italian 


" 


YB  23194 


